Research ArticleStatistical Inference in Dependent ComponentHybrid Systems with Masked Data

Naijun Sha1 Ronghua Wang2 Ping Hu2 and Xiaoling Xu3

1Department of Mathematical Sciences University of Texas at El Paso El Paso TX 79968 USA2College of Mathematics and Science Shanghai Normal University Shanghai 200234 China3Business Information Management School Shanghai University of International Business and Economics Shanghai 201600 China

Correspondence should be addressed to Naijun Sha nshautepedu

Received 26 March 2015 Accepted 26 May 2015

Academic Editor Shuo-Jye Wu

Copyright copy 2015 Naijun Sha et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Complex systems are usually composed of simple hybrid systems In this paper we consider statistical inference for two fundamentalhybrid systems series-parallel and parallel-series systems based on masked data Assuming dependent lifetimes of componentsmodelled by Marshall and Olkinrsquos bivariate exponential distribution in the system we present maximum likelihood and intervalestimation of parameters of interest Intensive simulation studies are performed to demonstrate the efficiency of the methods

1 Introduction

In a system consisting of several components the reliabilityanalyses are usually made by analyzing lifetime data Thesystem data includes two parts (i) the systemrsquos lifetime and(ii) the failure reason that is which component causes systemfailure In real situations however some things may preventsystems from revealing the failure reason such as shortageof funds limit of time error of records lack of diagnostictools and destructive consequences caused by the failure ofsome components For example in the reliability problemsof computers and integrated circuits the reason for thesystem failure is often attributed to a module containingseveral components but one could not determine exactlywhich component causes the system failure Therefore theobservable data from the test includes the failure time andfailure reason related to a subset of components In thesecases the reason for the failure of the system is masked andthe lifetime data is called masked data

The statistical analysis of masked data has a long historyUsher and Hodgson [1] initially proposed the parameterestimation under masked data Since then a significantamount of literature has emerged on various models In theseries system with constant linear and polynomial failurecomponents in the presence of masked data the maximum

likelihood (ML) and other estimation methods were stud-ied among many researchers (eg [2ndash6]) Sarhan and El-Bassiouny [7] considered a parallel systemusingmasked dataBayes methods with various priors were also used for theestimation of parameters in series and parallel systems (seeeg Sarhan [8ndash10] Jiang and Zhang [11]) El-Gohary [12]discussed a series system with two dependent componentsin a Bayesian approach So far most researches of maskeddata focused on a system with either series or parallel onlyand assumed independent and identical component lifetimein the system In many real situations however a ldquohybridrdquosystem is often seen in which the working components areconnected in a way of joining together with series andparallel For example currently air supply systems generallyare modular designed where the power system consists ofa number of semiconductor units combined in a series orhybrid method [13 14]

Complex systems are usually composed of simple sub-systems such as three-component series-parallel and parallel-series systems illustrated in Figure 1 In this paper we mainlyfocus on statistical inferences of the two fundamental hybridsystems in which the component lifetimes are nonindepen-dent and nonidentically distributed For the two systems firstwe note that the system failure occurrence is attributed toone of the four failures consisting of components 1 2 3 and

Hindawi Publishing CorporationAdvances in StatisticsVolume 2015 Article ID 525136 11 pageshttpdxdoiorg1011552015525136

2 Advances in Statistics

3

1

2

(a) Series-parallel system

1 2

3

(b) Parallel-series system

Figure 1 Hybrid systems of three components

12 where 12 denotes the occurrence of components 1 and2 failure simultaneously Let S be the set of all nine eventscausing the system failure that is

S = 1 2 3 12 1 2 1 3 2 3 12 3

1 2 3 (1)

If 119904 isin S consists of more than one element then thereason of the system failure is not exact and the life data ismasked Notice that here we differentiate two occurrences1 2 and 12 by assuming different independent processesdamaging component 1 only component 2 only and bothcomponents in the next sectionWemake statistical inferenceof parameters on likelihood-based methods in the presenceof masked data Section 2 presents the life distribution andreliability for the hybrid systems Section 3 concentrates onthe parameter estimation for the series-parallel and parallel-series systems respectively In Section 4 we assess theperformance of themethods on simulation studies Lastly weconclude the paper with a brief discussion in Section 5

2 Model Specification

For the three-component hybrid system in Figure 1 there is asubsystem consisting of components 1 and 2 From a practicalviewpoint the lifetimes of the components in the subsystemare usually dependent on each other and independent ofcomponent 3 outside the subsystem The unit lifetime modelis addressed in the following

21 Life Distribution A bivariate model is developed byMarshall andOlkin [15] to describe the correlated lifetimes oftwo units and is widely used in two-component system Basi-cally it was assumed that two-component system is affectedby ldquofatal shocksrdquo governed by three different independentPoisson processes with parameters 1205821 1205822 and 12058212 accordingto the shocks damage component 1 only component 2 onlyand both components respectively Particularly in the hybridsystem of Figure 1 the lifetimes 1198791 and 1198792 of the units 1and 2 are constructed through 119879

119894= min(119885

119894 11988512) 119894 = 1 2

where 1198851 1198852 and 11988512 are mutually independent randomvariables with 119885

119894sim exp(120582

119894) 119894 = 1 2 and 11988512 sim exp(12058212)

Then (1198791 1198792) follows a bivariate exponential distributionBiexp(1205821 1205822 12058212) whose joint reliability function is

11986512 (1199051 1199052) = exp minus12058211199051 minus12058221199052 minus12058212 max (1199051 1199052)

1199051 gt 0 1199052 gt 0(2)

and the joint density function is

11989112 (1199051 1199052)

=

1205822 (1205821 + 12058212) exp minus (1205821 + 12058212) 1199051 minus 12058221199052 1199051 gt 1199052

1205821 (1205822 + 12058212) exp minus (1205822 + 12058212) 1199052 minus 12058211199051 1199051 lt 1199052

(3)

The probability of both components failure at time 119905 cor-responds to the mass probability of singular part 11990112(119905) =

119875(1198791 = 1198792 = 119905) = 119875(1198851 ge 119905 1198852 ge 119905 11988512 = 119905) =

12058212119890minus(1205821+1205822+12058212)119905 119905 gt 0 The component 3 is shocked by

another independent Poisson process with parameter 120582 andso its lifetime is exponentially distributed with the density1198913(1199053) = 120582119890

minus1205821199053 1199053 gt 0 120582 gt 0

22 Reliability and Density of Hybrid System First we brieflyintroduce the concept of masked probability Assume thatthere is a masked event 119904 isin S with the exact failure case 119870

in the hybrid system then the probability of failure due tothe masked occurrence 119904 at time 119905 is

119875 (119905 lt119879le 119905 + 119889119905 119878 = 119904)

= sum

119895isin119904

119875 (119905 lt119879le 119905 + 119889119905 119878 = 119904 119870= 119895)

= sum

119895isin119904

119875 (119905 lt119879le 119905 + 119889119905 119870= 119895)

sdot 119875 (119878 = 119904 | 119905 lt 119879le 119905 + 119889119905 119870= 119895)

(4)

where 119875(119878 = 119904 | 119905 lt 119879 le 119905 + 119889119905 119870 = 119895) is called maskedprobability and 119875(119905 lt 119879 le 119905 + 119889119905 119870 = 119895) is the probabilityof system failure caused by component(s) 119895 at the time 119905 119895 =

1 2 3 12 In statistical analysis of masked data it is usuallyassumed that the masked occurrence is independent of thecause and failure time that is

119875 (119878 = 119904 | 119905 lt 119879le 119905 + 119889119905 119870= 119895) = 119875 (119878 = 119904) = 119888 (5)

The lifetime of the series-parallel system in Figure 1(a)is 119879 = min(max(1198791 1198792) 1198793) With the assumption that

Advances in Statistics 3

(1198791 1198792) sim Biexp(1205821 1205822 12058212) and an independent 1198793 sim

exp(120582) the reliability at time 119905 is

119875 (119879gt 119905)

= 119875 (max (1198791 1198792) gt 119905) 119875 (1198793 gt 119905)

= [1minus119875 (1198791 le 119905 1198792 le 119905)] 1198653 (119905)

= [1minus11986512 (119905 119905)] 1198653 (119905)

= [1minus∬

119905

011989112 (1199051 1199052) 11988911990511198891199052 minusint

119905

011990112 (119904) 119889119904] 1198653 (119905)

= 119890

minus(120582+1205821+1205822+12058212)119905(119890

1205821119905+ 119890

1205822119905minus 1) 119905 gt 0

(6)

and the probability densities of failure at time 119905 due to eachevent are

1198751 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 1)

= 119875 (119905 lt1198791 le 119905 + 119889119905 1198792 lt 119905) 119875 (1198793 gt 119905)

= int

119905

011989112 (119905 1199052) 1198891199052 times1198653 (119905)

= (1205821 +12058212) 119890minus(120582+1205821+12058212)119905

(1minus 119890

minus1205822119905)

1198752 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 2)

= 119875 (1198791 lt 119905 119905 lt 1198792 le 119905 + 119889119905) 119875 (1198793 gt 119905)

= int

119905

011989112 (1199051 119905) 1198891199051 times1198653 (119905)

= (1205822 +12058212) 119890minus(120582+1205822+12058212)119905

(1minus 119890

minus1205821119905)

11987512 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 12)

= 119875 (119905 lt1198791 =1198792 le 119905 + 119889119905) 119875 (1198793 gt 119905)

= 11990112 (119905) 1198653 (119905) = 12058212119890minus(120582+1205821+1205822+12058212)119905

1198753 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 3)

= [1minus119875 (1198791 le 119905 1198792 le 119905)] 1198913 (119905)

= [1minus11986512 (119905 119905)] 1198913 (119905)

= 120582119890

minus(120582+1205821+1205822+12058212)119905(119890

1205821119905+ 119890

1205822119905minus 1)

(7)

Likewise for the parallel-series system with three compo-nents as shown in Figure 1(b) the system life becomes 119879 =

max(min(1198791 1198792) 1198793) Therefore the reliability is

119875 (119879gt 119905) = 1minus119875 (min (1198791 1198792) le 119905 1198793 le 119905)

= 1minus [1minus119875 (1198791 gt 119905 1198792 gt 119905)] 1198653 (119905)

= 1minus [1minus11986512 (119905 119905)] 1198653 (119905)

= 119890

minus120582119905+ 119890

minus(1205821+1205822+12058212)119905minus 119890

minus(120582+1205821+1205822+12058212)119905

119905 gt 0

(8)

and the probability densities of failure at time 119905 due to eachcase are

1198751 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 1)

= 119875 (119905 lt1198791 le 119905 + 119889119905 1198792 gt 119905) 119875 (1198793 lt 119905)

= int

infin

119905

11989112 (119905 1199052) 1198891199052 times1198653 (119905)

= 1205821119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

1198752 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 2)

= 119875 (1198791 gt 119905 119905 lt 1198792 le 119905 + 119889119905) 119875 (1198793 lt 119905)

= int

infin

119905

11989112 (1199051 119905) 1198891199051 times1198653 (119905)

= 1205822119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

11987512 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 12)

= 119875 (119905 lt1198791 =1198792 le 119905 + 119889119905) 119875 (1198793 lt 119905)

= 11990112 (119905) 1198653 (119905) = 12058212119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

1198753 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 3)

= [1minus119875 (1198791 ge 119905 1198792 ge 119905)] 1198913 (119905)

= [1minus119875 (1198851 ge 119905 1198852 ge 119905 11988512 ge 119905)] 1198913 (119905)

= 120582119890

minus120582119905[1minus 119890

minus(1205821+1205822+12058212)119905]

(9)

Finally the density function for the system at 119905 due to themasked occurrence 119904 can be expressed as119875(119905 lt 119879 le 119905+119889119905 119878 =

119904) = sum119895isin119904

119888119875119895 The likelihood-based parameter inference for

the two hybrid systems is presented in the following

3 Parameter Estimation

In our statistical inference two common censoring schemesare considered type-I and type-II For 119899 tested systemsthrough reordering the failure times we assume thatthere are 119903

119896systems failures due to the 119896th mechanism

in S with the failure times 119905119903119896minus1+1 119905119903119896minus1+2 119905119903119896 where

119903119896

= sum

119896

119897=1 119903119897 1199030 = 0 119896 = 1 2 9 Obviously there

are totally 119903 = sum

9119897=1 119903119897

= 1199039 observed failure times and119899 minus 119903 censored observations For type-I censoring thetest is continuing until a prespecified time 120591 is reachedand we observed 119903 systems failed whereas for type-IIcensoring the test is carried out until the prespecified 119903

119896

systems failures for the 119896th mechanism and so the testterminated time 120591 = max(1199051 1199051199031 1199051199031+1 1199051199032 1199051199039)For both cases we express the observed life dataD = 1199051 1199051199031 1199051199031+1 1199051199032 1199051199039 120591 The correspondingmasked failure event is 119904

119894isin S for the system 119894 and masked

probability 119888119894

= 119875(119878119894

= 119904119894) 119894 = 1 2 119903 The probability

density of system 119894 at 119905119894for each case in (7) and (9) is

expressed as 119875119894119895

= 119875119895(119905119894) 119894 = 1 2 119903 119895 = 1 2 3 12

4 Advances in Statistics

indicating the failure due to the components 1 2 and 3 andboth components 1 and 2 respectively Finally the densityfunction for system 119894 at 119905

119894becomes sum

119895isin119904119894119888119894119875119894119895 Therefore

the applicable unified likelihood function for both hybridsystems and censoring schemes is

119871 (D) =

119903

prod

119894=1

sum

119895isin119904119894

119888119894119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

9prod

119896=1

119903119896

prod

119894=119903119896minus1+1

sum

119895isin119904119894

119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

1199031

prod

119894=11198751198941

1199032

prod

119894=1199031+11198751198942

1199033

prod

119894=1199032+11198751198943

1199034

prod

119894=1199033+111987511989412

1199035

prod

119894=1199034+1(1198751198941 +1198751198942)

sdot

1199036

prod

119894=1199035+1(1198751198941 +1198751198943)

1199037

prod

119894=1199036+1(1198751198942 +1198751198943)

1199038

prod

119894=1199037+1(11987511989412 +119875

1198943)

sdot

119903

prod

119894=1199038+1(1198751198941 +1198751198942 +1198751198943) [119875 (119879gt 120591)]

119899minus119903

(10)

where the constant 119862 = prod

119903

119894=1119888119894 does not contain theparameters of interest in 119875

119894119895

For the purpose of simplicity we only consider twospecial cases of failure rates (1) the components were shockedby independent Poisson processes with same parameters thatis 1205821 = 1205822 = 12058212 = 120582 (2) the Poisson processes affectingthe three components individually have the same parametersbut different from that of the Poisson process applying oncomponents 1 and 2 simultaneously that is 1205821 = 1205822 = 120582 =

12058212 The maximum likelihood estimation (MLE) approachwill be implemented for the inference To make notationsimpler we denote the log-likelihood function as 119897(120579) =

log 119871(120579 | D) where 120579 is the parameter of failure rates includedin the life densities We also apply the approximated chi-squared likelihood ratio statistic [16] to numerically obtainthe confidence intervals of parameters Particularly for ourcase the likelihood ratio statistic for the parameter Λ =

minus2log[119871(120579)119871(120579)] approximately follows 120594

2] where 120579 = 120582 or

120579 = (120582 12058212) and its MLE 120579 and ] is the dimension of 120579

In general this method works well even for the situationof small sample size that is the coverage probability of theconstructed interval is very close to the nominal confidencelevel

31 Series-Parallel System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (6) and thedensities in (7) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(2120582119890minus3120582119905119894 minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=+1199033+1(120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(4120582119890minus3120582119905119894 minus 4120582119890minus4120582119905119894)

sdot

1199037

prod

119894=1199035+1(4120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199037+1(2120582119890minus3120582119905119894)

sdot

119903

prod

119894=1199038+1(6120582119890minus3120582119905119894 minus 5120582119890minus4120582119905119894)

times 119890

minus4(119899minus119903)120582120591(2119890120582120591 minus 1)

119899minus119903

(11)

So the log-likelihood can be simplified as

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591 + (119899 minus 119903) log (2119890120582120591 minus 1)

+

119903

sum

119894=1log 120582minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (2119890120582119905119894 minus 2)

+

1199033

sum

119894=1199032+1log (2119890120582119905119894 minus 1) +

1199035

sum

119894=1199034+1log (4119890120582119905119894 minus 4)

+

1199037

sum

119894=1199035+1log (4119890120582119905119894 minus 3) +

1199038

sum

119894=1199037+1log (2119890120582119905119894)

+

119903

sum

119894=1199038+1log (6119890120582119905119894 minus 5)

(12)

and its derivative with respect to 120582 is

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 +

2 (119899 minus 119903) 120591

2 minus 119890

minus120582120591+

119903

120582

minus

119903

sum

119894=14119905119894

+

1199032

sum

119894=1

119905119894

1 minus 119890

minus120582119905119894+

1199033

sum

119894=1199032+1(

2119905119894

2 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

119905119894

1 minus 119890

minus120582119905119894)+

1199037

sum

119894=1199035+1(

4119905119894

4 minus 3119890minus120582119905119894)

+

1199038

sum

119894=1199037+1119905119894+

119903

sum

119894=1199038+1(

6119905119894

6 minus 5119890minus120582119905119894)

(13)

Since no analytical form of MLE 120582 can be obtained from

the equation 119897

1015840(120582) = 0 a numerical method has to be

implemented for specific data observations The uniqueness

Advances in Statistics 5

of MLE can be justified in the following way the termsinvolving exponent in 119897

1015840(120582) can be expressed as a unified

functional form 119892(120582) = 119886(119887 minus 119888119890

minus120582119905) with positive constants

119886 119887 and 119888 Since 119892

1015840(120582) = minus119886119888119905119890

minus120582119905(119887 minus 119888119890

minus120582119905)

2lt 0 we have

119897

10158401015840(120582) = minus

2 (119899 minus 119903) 120591

2119890

minus120582120591

(2 minus 119890

minus120582120591)

2 minus

119903

120582

2 minus

1199032

sum

119894=1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199033

sum

119894=1199032+1

21199052119894119890

minus120582119905119894

(2 minus 119890

minus120582119905119894)

2 minus

1199035

sum

119894=1199034+1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199037

sum

119894=1199035+1

121199052119894119890

minus120582119905119894

(4 minus 3119890minus120582119905119894)2minus

119903

sum

119894=1199038+1

301199052119894119890

minus120582119905119894

(6 minus 5119890minus120582119905119894)2

lt 0

(14)

Hence the log-likelihood function 119897(120582) is strictly concave andtherefore 119897

1015840(120582) = 0 implies a unique MLE

120582 Additionallylim120582rarr 01198971015840(120582) = infin lim

120582rarrinfin119897

1015840(120582) = minus3(119899 minus 119903)120591 minus 3sum

119903

119894=1 119905119894lt 0

and so the MLE 120582 is a positive value

(2) 1205821

= 1205822

= 120582 = 12058212 Under this case the likelihood

function (10) reduces to

119871 (120582 12058212 | D)

= 119862

1199032

prod

119894=1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(3120582+12058212)119905119894

]

sdot

1199035

prod

119894=1199034+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

1199038

prod

119894=1199037+1[12058212119890minus(3120582+12058212)119905119894

+120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

119903

prod

119894=1199038+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)] 119890

minus(119899minus119903)(2120582+12058212)120591(2

minus 119890

minus120582120591)

119899minus119903

(15)

and so the log-likelihood function is

119897 (120582 12058212)

= log119862minus (119899 minus 119903) (2120582+12058212) 120591

+ (119899 minus 119903) log (2minus 119890

minus120582120591) minus

119903

sum

119894=1(2120582+12058212) 119905119894

+

1199032

sum

119894=1[log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199033

sum

119894=1199032+1[log 120582+ log (2minus 119890

minus120582119905119894)]

+

1199034

sum

119894=1199033+1(log 12058212 minus120582119905

119894)

+

1199035

sum

119894=1199034+1[log 2+ log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [(3120582+12058212) minus (2120582+12058212) 119890

minus120582119905119894]

+

1199038

sum

119894=1199037+1log [2120582minus (120582 minus 12058212) 119890

minus120582119905119894]

+

119903

sum

119894=1199038+1log [(4120582+ 212058212) minus (3120582+ 212058212) 119890

minus120582119905119894]

(16)

TheMLEs 120582 12058212 can be obtained numerically in the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The existence of MLE isprovided in the Appendix

32 Parallel-Series System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (8) and thedensities in (9) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(120582119890

minus120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=1199033+1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199035+1(120582119890

minus120582119905119894+120582119890

minus3120582119905119894minus 2120582119890minus4120582119905119894)

6 Advances in Statistics

sdot

119903

prod

119894=1199038+1(120582119890

minus120582119905119894+ 2120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

times (119890

minus120582120591+ 119890

minus3120582120591minus 119890

minus4120582120591)

119899minus119903

(17)

and then the log-likelihood function is

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591

+ (119899 minus 119903) log (119890

3120582120591+ 119890

120582120591minus 1) +

119903

sum

119894=1log 120582

minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (119890

120582119905119894minus 1)

+

1199033

sum

119894=1199032+1log (119890

3120582119905119894minus 1) +

1199034

sum

119894=1199033+1log (119890

120582119905119894minus 1)

+

1199035

sum

119894=1199034+1log (2119890120582119905119894 minus 2)

+

1199038

sum

119894=1199035+1log (119890

3120582119905119894+ 119890

120582119905119894minus 2)

+

119903

sum

119894=1199038+1log (119890

3120582119905119894+ 2119890120582119905119894 minus 3)

(18)

Taking derivative with respect to 120582 we obtain

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 + (119899 minus 119903)

3120591 + 120591119890

minus2120582120591

1 + 119890

minus2120582120591minus 119890

minus3120582120591 +119903

120582

minus

119903

sum

119894=14119905119894+

1199032

sum

119894=1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

3119905119894

1 minus 119890

minus3120582119905119894)+

1199035

sum

119894=1199033+1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199038

sum

119894=1199035+1(

3119905119894+ 119905119894119890

minus2120582119905119894

1 + 119890

minus2120582119905119894minus 2119890minus3120582119905119894

)

+

119903

sum

119894=1199038+1(

3119905119894+ 2119905119894119890

minus2120582119905119894

1 + 2119890minus2120582119905119894 minus 3119890minus3120582119905119894)

(19)

Since lim120582rarr 01198971015840(120582) = infin and lim

120582rarrinfin119897

1015840(120582) = minus(119899 minus 119903)120591 minus

sum

119903

119894=1 4119905119894 + sum

1199032119894=1 119905119894+ sum

1199033119894=1199032+1

3119905119894+ sum

1199035119894=1199033+1

119905119894+ sum

119903

119894=1199035+1 3119905119894 lt 0119897

1015840(120582) = 0 has a positive root 120582

(2) 1205821= 1205822= 120582 = 120582

12 Under this special case the likelihood

function (10) then reduces to

119871 (120582 12058212 | D) = 119862

1199032

prod

119894=1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)]

sdot

1199035

prod

119894=1199034+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199038

prod

119894=1199037+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

119903

prod

119894=1199038+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)] (119890

minus120582120591+ 119890

minus(2120582+12058212)120591

minus 119890

minus(3120582+12058212)120591)

119899minus119903

(20)

and so the log-likelihood function is

119897 (120582 12058212) = log119862minus (119899 minus 119903) 120582120591 + (119899 minus 119903) log [1

+ 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591] minus

119903

sum

119894=1120582119905119894+

1199032

sum

119894=1[log 120582minus (120582

+ 12058212) 119905119894 + log (1minus 119890

minus120582119905119894)] +

1199033

sum

119894=1199032+1[log 120582

+ log (1minus 119890

minus(2120582+12058212)119905119894)] +

1199034

sum

119894=1199033+1[log 12058212 minus (120582 + 12058212)

sdot 119905119894+ log (1minus 119890

minus120582119905119894)] +

1199035

sum

119894=1199034+1[log 2minus (120582 + 12058212) 119905119894

+ log 120582+ log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [120582119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

Advances in Statistics 7

Table 1 Series-parallel system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10481 00267 07164 9835 4 3 3 3 2 2 1 1 10 10214 00226 06926 9814 4 4 2 2 2 2 2 2 08 08308 00177 05664 9845 4 3 3 3 2 2 1 1 08 08297 00172 05652 988

30

5 5 5 3 3 3 2 2 2 10 10371 00226 06366 9726 6 5 3 3 2 2 2 2 10 10288 00212 06196 9625 5 5 3 3 3 2 2 2 08 08357 00153 05116 9786 6 5 3 3 2 2 2 1 08 08255 00142 04956 972

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

3 3 3 2 2 2 2 2 1 10 10370 00298 07645 9824 4 3 2 2 2 2 1 1 10 10323 00283 07290 9813 3 3 2 2 2 2 2 1 08 08307 00175 06117 9854 4 3 2 2 2 2 1 1 08 08264 00172 05879 994

30

5 4 5 3 2 2 2 2 1 10 10407 00271 06749 9745 5 4 3 3 2 2 2 2 10 10380 00245 06491 9835 4 5 3 2 2 2 2 1 08 08325 00157 05368 9755 5 4 3 3 2 2 2 2 08 08245 00157 05193 983

+120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

1199038

sum

119894=1199037+1log [12058212119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

119903

sum

119894=1199038+1log [2120582119890minus(120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

(21)

The MLEs 120582 12058212 will be obtained numerically in the equa-

tions 120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The prove of theexistence of MLE is given in the Appendix

4 Simulation Study

In this section we conduct a simulation study to investi-gate the performance of our methodology We choose twoparameter values of failure rates for each case in the twohybrid systems that is 120582 = 10 08 in the case of samefailure rates and (120582 12058212) = (10 05) (08 04) for the case ofdifferent failure rates Under each setting of parameter valueswe carry out simulation study to generate the lifetimes 119879

119894

119894 = 1 2 3 following the construction described in Section 21under two sample sizes 119899 = 24 30 for each of which twocomplete samples (119899 = 119903 = sum

9119896=1 119903119896) with two settings

of failure numbers 119903119896and two censored samples with two

failure numbers (119903 = 20 21 for 119899 = 24 and 119903 = 26 28for 119899 = 30) are considered to determine the sample size119899 and 119903

119896variation effects for the estimation precision We

conduct 10000 Monte-Carlo simulations for each setting

of parameter value sample size and failure number Theaveraged MLE mean squared error (MSE) length of 95confidence interval and coverage probability are displayed inTables 1 and 2 for the series-parallel system and Tables 3 and4 for the parallel-series system

In each table the estimation results in the upper panelcorrespond to the complete sample and lower panel to thecensored sample It seems that the estimations are reasonablygood under these relative small sample sizes and all thecoverage probabilities of confidence intervals exceed thenominal confidence level indicating that it is a conservativemethod for interval estimation by chi-squared likelihoodratio statistics As expected under the same sample size119899 the MSEs and interval lengths are smaller in completesamples than these in censored samples Due to the scale ofthe true parameter values we noticed that given the samesample size 119899 and failure numbers 119903

119896rsquos the MSE and interval

length of estimates under larger true parameter values areconsistently larger than these under smaller true values InTable 1 for example given 119899 = 24 119903 = 24 the MSE =00267 and 95 confidence interval length = 07164 when120582 = 1 whereas MSE = 00177 and the length = 05664when 120582 = 08 However it is common that for a faircomparison between estimates variability with different unitsor different parameter values one should use a relativevariability measure such as coefficient of variation instead ofa measure of dispersion like MSE or interval length In ourcase we propose a ldquonormalizedrdquo measure of dispersion 119877 =

lengthestimate to remove the scale effect for the comparisonAs a result the estimation results mentioned above give us119877 = 0716410481 = 06818 and 0566408308 = 06817respectively which are very close to each other Similaroutcomes are obtained for other estimation results across thetables indicating a consistent precision for the estimationprocedure

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Statistics

3

1

2

(a) Series-parallel system

1 2

3

(b) Parallel-series system

Figure 1 Hybrid systems of three components

12 where 12 denotes the occurrence of components 1 and2 failure simultaneously Let S be the set of all nine eventscausing the system failure that is

S = 1 2 3 12 1 2 1 3 2 3 12 3

1 2 3 (1)

If 119904 isin S consists of more than one element then thereason of the system failure is not exact and the life data ismasked Notice that here we differentiate two occurrences1 2 and 12 by assuming different independent processesdamaging component 1 only component 2 only and bothcomponents in the next sectionWemake statistical inferenceof parameters on likelihood-based methods in the presenceof masked data Section 2 presents the life distribution andreliability for the hybrid systems Section 3 concentrates onthe parameter estimation for the series-parallel and parallel-series systems respectively In Section 4 we assess theperformance of themethods on simulation studies Lastly weconclude the paper with a brief discussion in Section 5

2 Model Specification

For the three-component hybrid system in Figure 1 there is asubsystem consisting of components 1 and 2 From a practicalviewpoint the lifetimes of the components in the subsystemare usually dependent on each other and independent ofcomponent 3 outside the subsystem The unit lifetime modelis addressed in the following

21 Life Distribution A bivariate model is developed byMarshall andOlkin [15] to describe the correlated lifetimes oftwo units and is widely used in two-component system Basi-cally it was assumed that two-component system is affectedby ldquofatal shocksrdquo governed by three different independentPoisson processes with parameters 1205821 1205822 and 12058212 accordingto the shocks damage component 1 only component 2 onlyand both components respectively Particularly in the hybridsystem of Figure 1 the lifetimes 1198791 and 1198792 of the units 1and 2 are constructed through 119879

119894= min(119885

119894 11988512) 119894 = 1 2

where 1198851 1198852 and 11988512 are mutually independent randomvariables with 119885

119894sim exp(120582

119894) 119894 = 1 2 and 11988512 sim exp(12058212)

Then (1198791 1198792) follows a bivariate exponential distributionBiexp(1205821 1205822 12058212) whose joint reliability function is

11986512 (1199051 1199052) = exp minus12058211199051 minus12058221199052 minus12058212 max (1199051 1199052)

1199051 gt 0 1199052 gt 0(2)

and the joint density function is

11989112 (1199051 1199052)

=

1205822 (1205821 + 12058212) exp minus (1205821 + 12058212) 1199051 minus 12058221199052 1199051 gt 1199052

1205821 (1205822 + 12058212) exp minus (1205822 + 12058212) 1199052 minus 12058211199051 1199051 lt 1199052

(3)

The probability of both components failure at time 119905 cor-responds to the mass probability of singular part 11990112(119905) =

119875(1198791 = 1198792 = 119905) = 119875(1198851 ge 119905 1198852 ge 119905 11988512 = 119905) =

12058212119890minus(1205821+1205822+12058212)119905 119905 gt 0 The component 3 is shocked by

another independent Poisson process with parameter 120582 andso its lifetime is exponentially distributed with the density1198913(1199053) = 120582119890

minus1205821199053 1199053 gt 0 120582 gt 0

22 Reliability and Density of Hybrid System First we brieflyintroduce the concept of masked probability Assume thatthere is a masked event 119904 isin S with the exact failure case 119870

in the hybrid system then the probability of failure due tothe masked occurrence 119904 at time 119905 is

119875 (119905 lt119879le 119905 + 119889119905 119878 = 119904)

= sum

119895isin119904

119875 (119905 lt119879le 119905 + 119889119905 119878 = 119904 119870= 119895)

= sum

119895isin119904

119875 (119905 lt119879le 119905 + 119889119905 119870= 119895)

sdot 119875 (119878 = 119904 | 119905 lt 119879le 119905 + 119889119905 119870= 119895)

(4)

where 119875(119878 = 119904 | 119905 lt 119879 le 119905 + 119889119905 119870 = 119895) is called maskedprobability and 119875(119905 lt 119879 le 119905 + 119889119905 119870 = 119895) is the probabilityof system failure caused by component(s) 119895 at the time 119905 119895 =

1 2 3 12 In statistical analysis of masked data it is usuallyassumed that the masked occurrence is independent of thecause and failure time that is

119875 (119878 = 119904 | 119905 lt 119879le 119905 + 119889119905 119870= 119895) = 119875 (119878 = 119904) = 119888 (5)

The lifetime of the series-parallel system in Figure 1(a)is 119879 = min(max(1198791 1198792) 1198793) With the assumption that

Advances in Statistics 3

(1198791 1198792) sim Biexp(1205821 1205822 12058212) and an independent 1198793 sim

exp(120582) the reliability at time 119905 is

119875 (119879gt 119905)

= 119875 (max (1198791 1198792) gt 119905) 119875 (1198793 gt 119905)

= [1minus119875 (1198791 le 119905 1198792 le 119905)] 1198653 (119905)

= [1minus11986512 (119905 119905)] 1198653 (119905)

= [1minus∬

119905

011989112 (1199051 1199052) 11988911990511198891199052 minusint

119905

011990112 (119904) 119889119904] 1198653 (119905)

= 119890

minus(120582+1205821+1205822+12058212)119905(119890

1205821119905+ 119890

1205822119905minus 1) 119905 gt 0

(6)

and the probability densities of failure at time 119905 due to eachevent are

1198751 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 1)

= 119875 (119905 lt1198791 le 119905 + 119889119905 1198792 lt 119905) 119875 (1198793 gt 119905)

= int

119905

011989112 (119905 1199052) 1198891199052 times1198653 (119905)

= (1205821 +12058212) 119890minus(120582+1205821+12058212)119905

(1minus 119890

minus1205822119905)

1198752 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 2)

= 119875 (1198791 lt 119905 119905 lt 1198792 le 119905 + 119889119905) 119875 (1198793 gt 119905)

= int

119905

011989112 (1199051 119905) 1198891199051 times1198653 (119905)

= (1205822 +12058212) 119890minus(120582+1205822+12058212)119905

(1minus 119890

minus1205821119905)

11987512 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 12)

= 119875 (119905 lt1198791 =1198792 le 119905 + 119889119905) 119875 (1198793 gt 119905)

= 11990112 (119905) 1198653 (119905) = 12058212119890minus(120582+1205821+1205822+12058212)119905

1198753 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 3)

= [1minus119875 (1198791 le 119905 1198792 le 119905)] 1198913 (119905)

= [1minus11986512 (119905 119905)] 1198913 (119905)

= 120582119890

minus(120582+1205821+1205822+12058212)119905(119890

1205821119905+ 119890

1205822119905minus 1)

(7)

Likewise for the parallel-series system with three compo-nents as shown in Figure 1(b) the system life becomes 119879 =

max(min(1198791 1198792) 1198793) Therefore the reliability is

119875 (119879gt 119905) = 1minus119875 (min (1198791 1198792) le 119905 1198793 le 119905)

= 1minus [1minus119875 (1198791 gt 119905 1198792 gt 119905)] 1198653 (119905)

= 1minus [1minus11986512 (119905 119905)] 1198653 (119905)

= 119890

minus120582119905+ 119890

minus(1205821+1205822+12058212)119905minus 119890

minus(120582+1205821+1205822+12058212)119905

119905 gt 0

(8)

and the probability densities of failure at time 119905 due to eachcase are

1198751 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 1)

= 119875 (119905 lt1198791 le 119905 + 119889119905 1198792 gt 119905) 119875 (1198793 lt 119905)

= int

infin

119905

11989112 (119905 1199052) 1198891199052 times1198653 (119905)

= 1205821119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

1198752 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 2)

= 119875 (1198791 gt 119905 119905 lt 1198792 le 119905 + 119889119905) 119875 (1198793 lt 119905)

= int

infin

119905

11989112 (1199051 119905) 1198891199051 times1198653 (119905)

= 1205822119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

11987512 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 12)

= 119875 (119905 lt1198791 =1198792 le 119905 + 119889119905) 119875 (1198793 lt 119905)

= 11990112 (119905) 1198653 (119905) = 12058212119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

1198753 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 3)

= [1minus119875 (1198791 ge 119905 1198792 ge 119905)] 1198913 (119905)

= [1minus119875 (1198851 ge 119905 1198852 ge 119905 11988512 ge 119905)] 1198913 (119905)

= 120582119890

minus120582119905[1minus 119890

minus(1205821+1205822+12058212)119905]

(9)

Finally the density function for the system at 119905 due to themasked occurrence 119904 can be expressed as119875(119905 lt 119879 le 119905+119889119905 119878 =

119904) = sum119895isin119904

119888119875119895 The likelihood-based parameter inference for

the two hybrid systems is presented in the following

3 Parameter Estimation

In our statistical inference two common censoring schemesare considered type-I and type-II For 119899 tested systemsthrough reordering the failure times we assume thatthere are 119903

119896systems failures due to the 119896th mechanism

in S with the failure times 119905119903119896minus1+1 119905119903119896minus1+2 119905119903119896 where

119903119896

= sum

119896

119897=1 119903119897 1199030 = 0 119896 = 1 2 9 Obviously there

are totally 119903 = sum

9119897=1 119903119897

= 1199039 observed failure times and119899 minus 119903 censored observations For type-I censoring thetest is continuing until a prespecified time 120591 is reachedand we observed 119903 systems failed whereas for type-IIcensoring the test is carried out until the prespecified 119903

119896

systems failures for the 119896th mechanism and so the testterminated time 120591 = max(1199051 1199051199031 1199051199031+1 1199051199032 1199051199039)For both cases we express the observed life dataD = 1199051 1199051199031 1199051199031+1 1199051199032 1199051199039 120591 The correspondingmasked failure event is 119904

119894isin S for the system 119894 and masked

probability 119888119894

= 119875(119878119894

= 119904119894) 119894 = 1 2 119903 The probability

density of system 119894 at 119905119894for each case in (7) and (9) is

expressed as 119875119894119895

= 119875119895(119905119894) 119894 = 1 2 119903 119895 = 1 2 3 12

4 Advances in Statistics

indicating the failure due to the components 1 2 and 3 andboth components 1 and 2 respectively Finally the densityfunction for system 119894 at 119905

119894becomes sum

119895isin119904119894119888119894119875119894119895 Therefore

the applicable unified likelihood function for both hybridsystems and censoring schemes is

119871 (D) =

119903

prod

119894=1

sum

119895isin119904119894

119888119894119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

9prod

119896=1

119903119896

prod

119894=119903119896minus1+1

sum

119895isin119904119894

119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

1199031

prod

119894=11198751198941

1199032

prod

119894=1199031+11198751198942

1199033

prod

119894=1199032+11198751198943

1199034

prod

119894=1199033+111987511989412

1199035

prod

119894=1199034+1(1198751198941 +1198751198942)

sdot

1199036

prod

119894=1199035+1(1198751198941 +1198751198943)

1199037

prod

119894=1199036+1(1198751198942 +1198751198943)

1199038

prod

119894=1199037+1(11987511989412 +119875

1198943)

sdot

119903

prod

119894=1199038+1(1198751198941 +1198751198942 +1198751198943) [119875 (119879gt 120591)]

119899minus119903

(10)

where the constant 119862 = prod

119903

119894=1119888119894 does not contain theparameters of interest in 119875

119894119895

For the purpose of simplicity we only consider twospecial cases of failure rates (1) the components were shockedby independent Poisson processes with same parameters thatis 1205821 = 1205822 = 12058212 = 120582 (2) the Poisson processes affectingthe three components individually have the same parametersbut different from that of the Poisson process applying oncomponents 1 and 2 simultaneously that is 1205821 = 1205822 = 120582 =

12058212 The maximum likelihood estimation (MLE) approachwill be implemented for the inference To make notationsimpler we denote the log-likelihood function as 119897(120579) =

log 119871(120579 | D) where 120579 is the parameter of failure rates includedin the life densities We also apply the approximated chi-squared likelihood ratio statistic [16] to numerically obtainthe confidence intervals of parameters Particularly for ourcase the likelihood ratio statistic for the parameter Λ =

minus2log[119871(120579)119871(120579)] approximately follows 120594

2] where 120579 = 120582 or

120579 = (120582 12058212) and its MLE 120579 and ] is the dimension of 120579

In general this method works well even for the situationof small sample size that is the coverage probability of theconstructed interval is very close to the nominal confidencelevel

31 Series-Parallel System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (6) and thedensities in (7) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(2120582119890minus3120582119905119894 minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=+1199033+1(120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(4120582119890minus3120582119905119894 minus 4120582119890minus4120582119905119894)

sdot

1199037

prod

119894=1199035+1(4120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199037+1(2120582119890minus3120582119905119894)

sdot

119903

prod

119894=1199038+1(6120582119890minus3120582119905119894 minus 5120582119890minus4120582119905119894)

times 119890

minus4(119899minus119903)120582120591(2119890120582120591 minus 1)

119899minus119903

(11)

So the log-likelihood can be simplified as

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591 + (119899 minus 119903) log (2119890120582120591 minus 1)

+

119903

sum

119894=1log 120582minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (2119890120582119905119894 minus 2)

+

1199033

sum

119894=1199032+1log (2119890120582119905119894 minus 1) +

1199035

sum

119894=1199034+1log (4119890120582119905119894 minus 4)

+

1199037

sum

119894=1199035+1log (4119890120582119905119894 minus 3) +

1199038

sum

119894=1199037+1log (2119890120582119905119894)

+

119903

sum

119894=1199038+1log (6119890120582119905119894 minus 5)

(12)

and its derivative with respect to 120582 is

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 +

2 (119899 minus 119903) 120591

2 minus 119890

minus120582120591+

119903

120582

minus

119903

sum

119894=14119905119894

+

1199032

sum

119894=1

119905119894

1 minus 119890

minus120582119905119894+

1199033

sum

119894=1199032+1(

2119905119894

2 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

119905119894

1 minus 119890

minus120582119905119894)+

1199037

sum

119894=1199035+1(

4119905119894

4 minus 3119890minus120582119905119894)

+

1199038

sum

119894=1199037+1119905119894+

119903

sum

119894=1199038+1(

6119905119894

6 minus 5119890minus120582119905119894)

(13)

Since no analytical form of MLE 120582 can be obtained from

the equation 119897

1015840(120582) = 0 a numerical method has to be

implemented for specific data observations The uniqueness

Advances in Statistics 5

of MLE can be justified in the following way the termsinvolving exponent in 119897

1015840(120582) can be expressed as a unified

functional form 119892(120582) = 119886(119887 minus 119888119890

minus120582119905) with positive constants

119886 119887 and 119888 Since 119892

1015840(120582) = minus119886119888119905119890

minus120582119905(119887 minus 119888119890

minus120582119905)

2lt 0 we have

119897

10158401015840(120582) = minus

2 (119899 minus 119903) 120591

2119890

minus120582120591

(2 minus 119890

minus120582120591)

2 minus

119903

120582

2 minus

1199032

sum

119894=1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199033

sum

119894=1199032+1

21199052119894119890

minus120582119905119894

(2 minus 119890

minus120582119905119894)

2 minus

1199035

sum

119894=1199034+1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199037

sum

119894=1199035+1

121199052119894119890

minus120582119905119894

(4 minus 3119890minus120582119905119894)2minus

119903

sum

119894=1199038+1

301199052119894119890

minus120582119905119894

(6 minus 5119890minus120582119905119894)2

lt 0

(14)

Hence the log-likelihood function 119897(120582) is strictly concave andtherefore 119897

1015840(120582) = 0 implies a unique MLE

120582 Additionallylim120582rarr 01198971015840(120582) = infin lim

120582rarrinfin119897

1015840(120582) = minus3(119899 minus 119903)120591 minus 3sum

119903

119894=1 119905119894lt 0

and so the MLE 120582 is a positive value

(2) 1205821

= 1205822

= 120582 = 12058212 Under this case the likelihood

function (10) reduces to

119871 (120582 12058212 | D)

= 119862

1199032

prod

119894=1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(3120582+12058212)119905119894

]

sdot

1199035

prod

119894=1199034+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

1199038

prod

119894=1199037+1[12058212119890minus(3120582+12058212)119905119894

+120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

119903

prod

119894=1199038+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)] 119890

minus(119899minus119903)(2120582+12058212)120591(2

minus 119890

minus120582120591)

119899minus119903

(15)

and so the log-likelihood function is

119897 (120582 12058212)

= log119862minus (119899 minus 119903) (2120582+12058212) 120591

+ (119899 minus 119903) log (2minus 119890

minus120582120591) minus

119903

sum

119894=1(2120582+12058212) 119905119894

+

1199032

sum

119894=1[log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199033

sum

119894=1199032+1[log 120582+ log (2minus 119890

minus120582119905119894)]

+

1199034

sum

119894=1199033+1(log 12058212 minus120582119905

119894)

+

1199035

sum

119894=1199034+1[log 2+ log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [(3120582+12058212) minus (2120582+12058212) 119890

minus120582119905119894]

+

1199038

sum

119894=1199037+1log [2120582minus (120582 minus 12058212) 119890

minus120582119905119894]

+

119903

sum

119894=1199038+1log [(4120582+ 212058212) minus (3120582+ 212058212) 119890

minus120582119905119894]

(16)

TheMLEs 120582 12058212 can be obtained numerically in the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The existence of MLE isprovided in the Appendix

32 Parallel-Series System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (8) and thedensities in (9) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(120582119890

minus120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=1199033+1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199035+1(120582119890

minus120582119905119894+120582119890

minus3120582119905119894minus 2120582119890minus4120582119905119894)

6 Advances in Statistics

sdot

119903

prod

119894=1199038+1(120582119890

minus120582119905119894+ 2120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

times (119890

minus120582120591+ 119890

minus3120582120591minus 119890

minus4120582120591)

119899minus119903

(17)

and then the log-likelihood function is

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591

+ (119899 minus 119903) log (119890

3120582120591+ 119890

120582120591minus 1) +

119903

sum

119894=1log 120582

minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (119890

120582119905119894minus 1)

+

1199033

sum

119894=1199032+1log (119890

3120582119905119894minus 1) +

1199034

sum

119894=1199033+1log (119890

120582119905119894minus 1)

+

1199035

sum

119894=1199034+1log (2119890120582119905119894 minus 2)

+

1199038

sum

119894=1199035+1log (119890

3120582119905119894+ 119890

120582119905119894minus 2)

+

119903

sum

119894=1199038+1log (119890

3120582119905119894+ 2119890120582119905119894 minus 3)

(18)

Taking derivative with respect to 120582 we obtain

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 + (119899 minus 119903)

3120591 + 120591119890

minus2120582120591

1 + 119890

minus2120582120591minus 119890

minus3120582120591 +119903

120582

minus

119903

sum

119894=14119905119894+

1199032

sum

119894=1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

3119905119894

1 minus 119890

minus3120582119905119894)+

1199035

sum

119894=1199033+1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199038

sum

119894=1199035+1(

3119905119894+ 119905119894119890

minus2120582119905119894

1 + 119890

minus2120582119905119894minus 2119890minus3120582119905119894

)

+

119903

sum

119894=1199038+1(

3119905119894+ 2119905119894119890

minus2120582119905119894

1 + 2119890minus2120582119905119894 minus 3119890minus3120582119905119894)

(19)

Since lim120582rarr 01198971015840(120582) = infin and lim

120582rarrinfin119897

1015840(120582) = minus(119899 minus 119903)120591 minus

sum

119903

119894=1 4119905119894 + sum

1199032119894=1 119905119894+ sum

1199033119894=1199032+1

3119905119894+ sum

1199035119894=1199033+1

119905119894+ sum

119903

119894=1199035+1 3119905119894 lt 0119897

1015840(120582) = 0 has a positive root 120582

(2) 1205821= 1205822= 120582 = 120582

12 Under this special case the likelihood

function (10) then reduces to

119871 (120582 12058212 | D) = 119862

1199032

prod

119894=1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)]

sdot

1199035

prod

119894=1199034+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199038

prod

119894=1199037+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

119903

prod

119894=1199038+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)] (119890

minus120582120591+ 119890

minus(2120582+12058212)120591

minus 119890

minus(3120582+12058212)120591)

119899minus119903

(20)

and so the log-likelihood function is

119897 (120582 12058212) = log119862minus (119899 minus 119903) 120582120591 + (119899 minus 119903) log [1

+ 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591] minus

119903

sum

119894=1120582119905119894+

1199032

sum

119894=1[log 120582minus (120582

+ 12058212) 119905119894 + log (1minus 119890

minus120582119905119894)] +

1199033

sum

119894=1199032+1[log 120582

+ log (1minus 119890

minus(2120582+12058212)119905119894)] +

1199034

sum

119894=1199033+1[log 12058212 minus (120582 + 12058212)

sdot 119905119894+ log (1minus 119890

minus120582119905119894)] +

1199035

sum

119894=1199034+1[log 2minus (120582 + 12058212) 119905119894

+ log 120582+ log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [120582119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

Advances in Statistics 7

Table 1 Series-parallel system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10481 00267 07164 9835 4 3 3 3 2 2 1 1 10 10214 00226 06926 9814 4 4 2 2 2 2 2 2 08 08308 00177 05664 9845 4 3 3 3 2 2 1 1 08 08297 00172 05652 988

30

5 5 5 3 3 3 2 2 2 10 10371 00226 06366 9726 6 5 3 3 2 2 2 2 10 10288 00212 06196 9625 5 5 3 3 3 2 2 2 08 08357 00153 05116 9786 6 5 3 3 2 2 2 1 08 08255 00142 04956 972

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

3 3 3 2 2 2 2 2 1 10 10370 00298 07645 9824 4 3 2 2 2 2 1 1 10 10323 00283 07290 9813 3 3 2 2 2 2 2 1 08 08307 00175 06117 9854 4 3 2 2 2 2 1 1 08 08264 00172 05879 994

30

5 4 5 3 2 2 2 2 1 10 10407 00271 06749 9745 5 4 3 3 2 2 2 2 10 10380 00245 06491 9835 4 5 3 2 2 2 2 1 08 08325 00157 05368 9755 5 4 3 3 2 2 2 2 08 08245 00157 05193 983

+120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

1199038

sum

119894=1199037+1log [12058212119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

119903

sum

119894=1199038+1log [2120582119890minus(120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

(21)

The MLEs 120582 12058212 will be obtained numerically in the equa-

tions 120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The prove of theexistence of MLE is given in the Appendix

4 Simulation Study

In this section we conduct a simulation study to investi-gate the performance of our methodology We choose twoparameter values of failure rates for each case in the twohybrid systems that is 120582 = 10 08 in the case of samefailure rates and (120582 12058212) = (10 05) (08 04) for the case ofdifferent failure rates Under each setting of parameter valueswe carry out simulation study to generate the lifetimes 119879

119894

119894 = 1 2 3 following the construction described in Section 21under two sample sizes 119899 = 24 30 for each of which twocomplete samples (119899 = 119903 = sum

9119896=1 119903119896) with two settings

of failure numbers 119903119896and two censored samples with two

failure numbers (119903 = 20 21 for 119899 = 24 and 119903 = 26 28for 119899 = 30) are considered to determine the sample size119899 and 119903

119896variation effects for the estimation precision We

conduct 10000 Monte-Carlo simulations for each setting

of parameter value sample size and failure number Theaveraged MLE mean squared error (MSE) length of 95confidence interval and coverage probability are displayed inTables 1 and 2 for the series-parallel system and Tables 3 and4 for the parallel-series system

In each table the estimation results in the upper panelcorrespond to the complete sample and lower panel to thecensored sample It seems that the estimations are reasonablygood under these relative small sample sizes and all thecoverage probabilities of confidence intervals exceed thenominal confidence level indicating that it is a conservativemethod for interval estimation by chi-squared likelihoodratio statistics As expected under the same sample size119899 the MSEs and interval lengths are smaller in completesamples than these in censored samples Due to the scale ofthe true parameter values we noticed that given the samesample size 119899 and failure numbers 119903

119896rsquos the MSE and interval

length of estimates under larger true parameter values areconsistently larger than these under smaller true values InTable 1 for example given 119899 = 24 119903 = 24 the MSE =00267 and 95 confidence interval length = 07164 when120582 = 1 whereas MSE = 00177 and the length = 05664when 120582 = 08 However it is common that for a faircomparison between estimates variability with different unitsor different parameter values one should use a relativevariability measure such as coefficient of variation instead ofa measure of dispersion like MSE or interval length In ourcase we propose a ldquonormalizedrdquo measure of dispersion 119877 =

lengthestimate to remove the scale effect for the comparisonAs a result the estimation results mentioned above give us119877 = 0716410481 = 06818 and 0566408308 = 06817respectively which are very close to each other Similaroutcomes are obtained for other estimation results across thetables indicating a consistent precision for the estimationprocedure

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 3

(1198791 1198792) sim Biexp(1205821 1205822 12058212) and an independent 1198793 sim

exp(120582) the reliability at time 119905 is

119875 (119879gt 119905)

= 119875 (max (1198791 1198792) gt 119905) 119875 (1198793 gt 119905)

= [1minus119875 (1198791 le 119905 1198792 le 119905)] 1198653 (119905)

= [1minus11986512 (119905 119905)] 1198653 (119905)

= [1minus∬

119905

011989112 (1199051 1199052) 11988911990511198891199052 minusint

119905

011990112 (119904) 119889119904] 1198653 (119905)

= 119890

minus(120582+1205821+1205822+12058212)119905(119890

1205821119905+ 119890

1205822119905minus 1) 119905 gt 0

(6)

and the probability densities of failure at time 119905 due to eachevent are

1198751 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 1)

= 119875 (119905 lt1198791 le 119905 + 119889119905 1198792 lt 119905) 119875 (1198793 gt 119905)

= int

119905

011989112 (119905 1199052) 1198891199052 times1198653 (119905)

= (1205821 +12058212) 119890minus(120582+1205821+12058212)119905

(1minus 119890

minus1205822119905)

1198752 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 2)

= 119875 (1198791 lt 119905 119905 lt 1198792 le 119905 + 119889119905) 119875 (1198793 gt 119905)

= int

119905

011989112 (1199051 119905) 1198891199051 times1198653 (119905)

= (1205822 +12058212) 119890minus(120582+1205822+12058212)119905

(1minus 119890

minus1205821119905)

11987512 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 12)

= 119875 (119905 lt1198791 =1198792 le 119905 + 119889119905) 119875 (1198793 gt 119905)

= 11990112 (119905) 1198653 (119905) = 12058212119890minus(120582+1205821+1205822+12058212)119905

1198753 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 3)

= [1minus119875 (1198791 le 119905 1198792 le 119905)] 1198913 (119905)

= [1minus11986512 (119905 119905)] 1198913 (119905)

= 120582119890

minus(120582+1205821+1205822+12058212)119905(119890

1205821119905+ 119890

1205822119905minus 1)

(7)

Likewise for the parallel-series system with three compo-nents as shown in Figure 1(b) the system life becomes 119879 =

max(min(1198791 1198792) 1198793) Therefore the reliability is

119875 (119879gt 119905) = 1minus119875 (min (1198791 1198792) le 119905 1198793 le 119905)

= 1minus [1minus119875 (1198791 gt 119905 1198792 gt 119905)] 1198653 (119905)

= 1minus [1minus11986512 (119905 119905)] 1198653 (119905)

= 119890

minus120582119905+ 119890

minus(1205821+1205822+12058212)119905minus 119890

minus(120582+1205821+1205822+12058212)119905

119905 gt 0

(8)

and the probability densities of failure at time 119905 due to eachcase are

1198751 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 1)

= 119875 (119905 lt1198791 le 119905 + 119889119905 1198792 gt 119905) 119875 (1198793 lt 119905)

= int

infin

119905

11989112 (119905 1199052) 1198891199052 times1198653 (119905)

= 1205821119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

1198752 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 2)

= 119875 (1198791 gt 119905 119905 lt 1198792 le 119905 + 119889119905) 119875 (1198793 lt 119905)

= int

infin

119905

11989112 (1199051 119905) 1198891199051 times1198653 (119905)

= 1205822119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

11987512 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 12)

= 119875 (119905 lt1198791 =1198792 le 119905 + 119889119905) 119875 (1198793 lt 119905)

= 11990112 (119905) 1198653 (119905) = 12058212119890minus(1205821+1205822+12058212)119905

(1minus 119890

minus120582119905)

1198753 = 119875 (119905 lt119879le 119905 + 119889119905 119870= 3)

= [1minus119875 (1198791 ge 119905 1198792 ge 119905)] 1198913 (119905)

= [1minus119875 (1198851 ge 119905 1198852 ge 119905 11988512 ge 119905)] 1198913 (119905)

= 120582119890

minus120582119905[1minus 119890

minus(1205821+1205822+12058212)119905]

(9)

Finally the density function for the system at 119905 due to themasked occurrence 119904 can be expressed as119875(119905 lt 119879 le 119905+119889119905 119878 =

119904) = sum119895isin119904

119888119875119895 The likelihood-based parameter inference for

the two hybrid systems is presented in the following

3 Parameter Estimation

In our statistical inference two common censoring schemesare considered type-I and type-II For 119899 tested systemsthrough reordering the failure times we assume thatthere are 119903

119896systems failures due to the 119896th mechanism

in S with the failure times 119905119903119896minus1+1 119905119903119896minus1+2 119905119903119896 where

119903119896

= sum

119896

119897=1 119903119897 1199030 = 0 119896 = 1 2 9 Obviously there

are totally 119903 = sum

9119897=1 119903119897

= 1199039 observed failure times and119899 minus 119903 censored observations For type-I censoring thetest is continuing until a prespecified time 120591 is reachedand we observed 119903 systems failed whereas for type-IIcensoring the test is carried out until the prespecified 119903

119896

systems failures for the 119896th mechanism and so the testterminated time 120591 = max(1199051 1199051199031 1199051199031+1 1199051199032 1199051199039)For both cases we express the observed life dataD = 1199051 1199051199031 1199051199031+1 1199051199032 1199051199039 120591 The correspondingmasked failure event is 119904

119894isin S for the system 119894 and masked

probability 119888119894

= 119875(119878119894

= 119904119894) 119894 = 1 2 119903 The probability

density of system 119894 at 119905119894for each case in (7) and (9) is

expressed as 119875119894119895

= 119875119895(119905119894) 119894 = 1 2 119903 119895 = 1 2 3 12

4 Advances in Statistics

indicating the failure due to the components 1 2 and 3 andboth components 1 and 2 respectively Finally the densityfunction for system 119894 at 119905

119894becomes sum

119895isin119904119894119888119894119875119894119895 Therefore

the applicable unified likelihood function for both hybridsystems and censoring schemes is

119871 (D) =

119903

prod

119894=1

sum

119895isin119904119894

119888119894119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

9prod

119896=1

119903119896

prod

119894=119903119896minus1+1

sum

119895isin119904119894

119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

1199031

prod

119894=11198751198941

1199032

prod

119894=1199031+11198751198942

1199033

prod

119894=1199032+11198751198943

1199034

prod

119894=1199033+111987511989412

1199035

prod

119894=1199034+1(1198751198941 +1198751198942)

sdot

1199036

prod

119894=1199035+1(1198751198941 +1198751198943)

1199037

prod

119894=1199036+1(1198751198942 +1198751198943)

1199038

prod

119894=1199037+1(11987511989412 +119875

1198943)

sdot

119903

prod

119894=1199038+1(1198751198941 +1198751198942 +1198751198943) [119875 (119879gt 120591)]

119899minus119903

(10)

where the constant 119862 = prod

119903

119894=1119888119894 does not contain theparameters of interest in 119875

119894119895

For the purpose of simplicity we only consider twospecial cases of failure rates (1) the components were shockedby independent Poisson processes with same parameters thatis 1205821 = 1205822 = 12058212 = 120582 (2) the Poisson processes affectingthe three components individually have the same parametersbut different from that of the Poisson process applying oncomponents 1 and 2 simultaneously that is 1205821 = 1205822 = 120582 =

12058212 The maximum likelihood estimation (MLE) approachwill be implemented for the inference To make notationsimpler we denote the log-likelihood function as 119897(120579) =

log 119871(120579 | D) where 120579 is the parameter of failure rates includedin the life densities We also apply the approximated chi-squared likelihood ratio statistic [16] to numerically obtainthe confidence intervals of parameters Particularly for ourcase the likelihood ratio statistic for the parameter Λ =

minus2log[119871(120579)119871(120579)] approximately follows 120594

2] where 120579 = 120582 or

120579 = (120582 12058212) and its MLE 120579 and ] is the dimension of 120579

In general this method works well even for the situationof small sample size that is the coverage probability of theconstructed interval is very close to the nominal confidencelevel

31 Series-Parallel System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (6) and thedensities in (7) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(2120582119890minus3120582119905119894 minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=+1199033+1(120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(4120582119890minus3120582119905119894 minus 4120582119890minus4120582119905119894)

sdot

1199037

prod

119894=1199035+1(4120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199037+1(2120582119890minus3120582119905119894)

sdot

119903

prod

119894=1199038+1(6120582119890minus3120582119905119894 minus 5120582119890minus4120582119905119894)

times 119890

minus4(119899minus119903)120582120591(2119890120582120591 minus 1)

119899minus119903

(11)

So the log-likelihood can be simplified as

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591 + (119899 minus 119903) log (2119890120582120591 minus 1)

+

119903

sum

119894=1log 120582minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (2119890120582119905119894 minus 2)

+

1199033

sum

119894=1199032+1log (2119890120582119905119894 minus 1) +

1199035

sum

119894=1199034+1log (4119890120582119905119894 minus 4)

+

1199037

sum

119894=1199035+1log (4119890120582119905119894 minus 3) +

1199038

sum

119894=1199037+1log (2119890120582119905119894)

+

119903

sum

119894=1199038+1log (6119890120582119905119894 minus 5)

(12)

and its derivative with respect to 120582 is

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 +

2 (119899 minus 119903) 120591

2 minus 119890

minus120582120591+

119903

120582

minus

119903

sum

119894=14119905119894

+

1199032

sum

119894=1

119905119894

1 minus 119890

minus120582119905119894+

1199033

sum

119894=1199032+1(

2119905119894

2 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

119905119894

1 minus 119890

minus120582119905119894)+

1199037

sum

119894=1199035+1(

4119905119894

4 minus 3119890minus120582119905119894)

+

1199038

sum

119894=1199037+1119905119894+

119903

sum

119894=1199038+1(

6119905119894

6 minus 5119890minus120582119905119894)

(13)

Since no analytical form of MLE 120582 can be obtained from

the equation 119897

1015840(120582) = 0 a numerical method has to be

implemented for specific data observations The uniqueness

Advances in Statistics 5

of MLE can be justified in the following way the termsinvolving exponent in 119897

1015840(120582) can be expressed as a unified

functional form 119892(120582) = 119886(119887 minus 119888119890

minus120582119905) with positive constants

119886 119887 and 119888 Since 119892

1015840(120582) = minus119886119888119905119890

minus120582119905(119887 minus 119888119890

minus120582119905)

2lt 0 we have

119897

10158401015840(120582) = minus

2 (119899 minus 119903) 120591

2119890

minus120582120591

(2 minus 119890

minus120582120591)

2 minus

119903

120582

2 minus

1199032

sum

119894=1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199033

sum

119894=1199032+1

21199052119894119890

minus120582119905119894

(2 minus 119890

minus120582119905119894)

2 minus

1199035

sum

119894=1199034+1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199037

sum

119894=1199035+1

121199052119894119890

minus120582119905119894

(4 minus 3119890minus120582119905119894)2minus

119903

sum

119894=1199038+1

301199052119894119890

minus120582119905119894

(6 minus 5119890minus120582119905119894)2

lt 0

(14)

Hence the log-likelihood function 119897(120582) is strictly concave andtherefore 119897

1015840(120582) = 0 implies a unique MLE

120582 Additionallylim120582rarr 01198971015840(120582) = infin lim

120582rarrinfin119897

1015840(120582) = minus3(119899 minus 119903)120591 minus 3sum

119903

119894=1 119905119894lt 0

and so the MLE 120582 is a positive value

(2) 1205821

= 1205822

= 120582 = 12058212 Under this case the likelihood

function (10) reduces to

119871 (120582 12058212 | D)

= 119862

1199032

prod

119894=1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(3120582+12058212)119905119894

]

sdot

1199035

prod

119894=1199034+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

1199038

prod

119894=1199037+1[12058212119890minus(3120582+12058212)119905119894

+120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

119903

prod

119894=1199038+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)] 119890

minus(119899minus119903)(2120582+12058212)120591(2

minus 119890

minus120582120591)

119899minus119903

(15)

and so the log-likelihood function is

119897 (120582 12058212)

= log119862minus (119899 minus 119903) (2120582+12058212) 120591

+ (119899 minus 119903) log (2minus 119890

minus120582120591) minus

119903

sum

119894=1(2120582+12058212) 119905119894

+

1199032

sum

119894=1[log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199033

sum

119894=1199032+1[log 120582+ log (2minus 119890

minus120582119905119894)]

+

1199034

sum

119894=1199033+1(log 12058212 minus120582119905

119894)

+

1199035

sum

119894=1199034+1[log 2+ log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [(3120582+12058212) minus (2120582+12058212) 119890

minus120582119905119894]

+

1199038

sum

119894=1199037+1log [2120582minus (120582 minus 12058212) 119890

minus120582119905119894]

+

119903

sum

119894=1199038+1log [(4120582+ 212058212) minus (3120582+ 212058212) 119890

minus120582119905119894]

(16)

TheMLEs 120582 12058212 can be obtained numerically in the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The existence of MLE isprovided in the Appendix

32 Parallel-Series System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (8) and thedensities in (9) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(120582119890

minus120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=1199033+1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199035+1(120582119890

minus120582119905119894+120582119890

minus3120582119905119894minus 2120582119890minus4120582119905119894)

6 Advances in Statistics

sdot

119903

prod

119894=1199038+1(120582119890

minus120582119905119894+ 2120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

times (119890

minus120582120591+ 119890

minus3120582120591minus 119890

minus4120582120591)

119899minus119903

(17)

and then the log-likelihood function is

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591

+ (119899 minus 119903) log (119890

3120582120591+ 119890

120582120591minus 1) +

119903

sum

119894=1log 120582

minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (119890

120582119905119894minus 1)

+

1199033

sum

119894=1199032+1log (119890

3120582119905119894minus 1) +

1199034

sum

119894=1199033+1log (119890

120582119905119894minus 1)

+

1199035

sum

119894=1199034+1log (2119890120582119905119894 minus 2)

+

1199038

sum

119894=1199035+1log (119890

3120582119905119894+ 119890

120582119905119894minus 2)

+

119903

sum

119894=1199038+1log (119890

3120582119905119894+ 2119890120582119905119894 minus 3)

(18)

Taking derivative with respect to 120582 we obtain

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 + (119899 minus 119903)

3120591 + 120591119890

minus2120582120591

1 + 119890

minus2120582120591minus 119890

minus3120582120591 +119903

120582

minus

119903

sum

119894=14119905119894+

1199032

sum

119894=1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

3119905119894

1 minus 119890

minus3120582119905119894)+

1199035

sum

119894=1199033+1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199038

sum

119894=1199035+1(

3119905119894+ 119905119894119890

minus2120582119905119894

1 + 119890

minus2120582119905119894minus 2119890minus3120582119905119894

)

+

119903

sum

119894=1199038+1(

3119905119894+ 2119905119894119890

minus2120582119905119894

1 + 2119890minus2120582119905119894 minus 3119890minus3120582119905119894)

(19)

Since lim120582rarr 01198971015840(120582) = infin and lim

120582rarrinfin119897

1015840(120582) = minus(119899 minus 119903)120591 minus

sum

119903

119894=1 4119905119894 + sum

1199032119894=1 119905119894+ sum

1199033119894=1199032+1

3119905119894+ sum

1199035119894=1199033+1

119905119894+ sum

119903

119894=1199035+1 3119905119894 lt 0119897

1015840(120582) = 0 has a positive root 120582

(2) 1205821= 1205822= 120582 = 120582

12 Under this special case the likelihood

function (10) then reduces to

119871 (120582 12058212 | D) = 119862

1199032

prod

119894=1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)]

sdot

1199035

prod

119894=1199034+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199038

prod

119894=1199037+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

119903

prod

119894=1199038+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)] (119890

minus120582120591+ 119890

minus(2120582+12058212)120591

minus 119890

minus(3120582+12058212)120591)

119899minus119903

(20)

and so the log-likelihood function is

119897 (120582 12058212) = log119862minus (119899 minus 119903) 120582120591 + (119899 minus 119903) log [1

+ 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591] minus

119903

sum

119894=1120582119905119894+

1199032

sum

119894=1[log 120582minus (120582

+ 12058212) 119905119894 + log (1minus 119890

minus120582119905119894)] +

1199033

sum

119894=1199032+1[log 120582

+ log (1minus 119890

minus(2120582+12058212)119905119894)] +

1199034

sum

119894=1199033+1[log 12058212 minus (120582 + 12058212)

sdot 119905119894+ log (1minus 119890

minus120582119905119894)] +

1199035

sum

119894=1199034+1[log 2minus (120582 + 12058212) 119905119894

+ log 120582+ log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [120582119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

Advances in Statistics 7

Table 1 Series-parallel system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10481 00267 07164 9835 4 3 3 3 2 2 1 1 10 10214 00226 06926 9814 4 4 2 2 2 2 2 2 08 08308 00177 05664 9845 4 3 3 3 2 2 1 1 08 08297 00172 05652 988

30

5 5 5 3 3 3 2 2 2 10 10371 00226 06366 9726 6 5 3 3 2 2 2 2 10 10288 00212 06196 9625 5 5 3 3 3 2 2 2 08 08357 00153 05116 9786 6 5 3 3 2 2 2 1 08 08255 00142 04956 972

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

3 3 3 2 2 2 2 2 1 10 10370 00298 07645 9824 4 3 2 2 2 2 1 1 10 10323 00283 07290 9813 3 3 2 2 2 2 2 1 08 08307 00175 06117 9854 4 3 2 2 2 2 1 1 08 08264 00172 05879 994

30

5 4 5 3 2 2 2 2 1 10 10407 00271 06749 9745 5 4 3 3 2 2 2 2 10 10380 00245 06491 9835 4 5 3 2 2 2 2 1 08 08325 00157 05368 9755 5 4 3 3 2 2 2 2 08 08245 00157 05193 983

+120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

1199038

sum

119894=1199037+1log [12058212119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

119903

sum

119894=1199038+1log [2120582119890minus(120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

(21)

The MLEs 120582 12058212 will be obtained numerically in the equa-

tions 120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The prove of theexistence of MLE is given in the Appendix

4 Simulation Study

In this section we conduct a simulation study to investi-gate the performance of our methodology We choose twoparameter values of failure rates for each case in the twohybrid systems that is 120582 = 10 08 in the case of samefailure rates and (120582 12058212) = (10 05) (08 04) for the case ofdifferent failure rates Under each setting of parameter valueswe carry out simulation study to generate the lifetimes 119879

119894

119894 = 1 2 3 following the construction described in Section 21under two sample sizes 119899 = 24 30 for each of which twocomplete samples (119899 = 119903 = sum

9119896=1 119903119896) with two settings

of failure numbers 119903119896and two censored samples with two

failure numbers (119903 = 20 21 for 119899 = 24 and 119903 = 26 28for 119899 = 30) are considered to determine the sample size119899 and 119903

119896variation effects for the estimation precision We

conduct 10000 Monte-Carlo simulations for each setting

of parameter value sample size and failure number Theaveraged MLE mean squared error (MSE) length of 95confidence interval and coverage probability are displayed inTables 1 and 2 for the series-parallel system and Tables 3 and4 for the parallel-series system

In each table the estimation results in the upper panelcorrespond to the complete sample and lower panel to thecensored sample It seems that the estimations are reasonablygood under these relative small sample sizes and all thecoverage probabilities of confidence intervals exceed thenominal confidence level indicating that it is a conservativemethod for interval estimation by chi-squared likelihoodratio statistics As expected under the same sample size119899 the MSEs and interval lengths are smaller in completesamples than these in censored samples Due to the scale ofthe true parameter values we noticed that given the samesample size 119899 and failure numbers 119903

119896rsquos the MSE and interval

length of estimates under larger true parameter values areconsistently larger than these under smaller true values InTable 1 for example given 119899 = 24 119903 = 24 the MSE =00267 and 95 confidence interval length = 07164 when120582 = 1 whereas MSE = 00177 and the length = 05664when 120582 = 08 However it is common that for a faircomparison between estimates variability with different unitsor different parameter values one should use a relativevariability measure such as coefficient of variation instead ofa measure of dispersion like MSE or interval length In ourcase we propose a ldquonormalizedrdquo measure of dispersion 119877 =

lengthestimate to remove the scale effect for the comparisonAs a result the estimation results mentioned above give us119877 = 0716410481 = 06818 and 0566408308 = 06817respectively which are very close to each other Similaroutcomes are obtained for other estimation results across thetables indicating a consistent precision for the estimationprocedure

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Statistics

indicating the failure due to the components 1 2 and 3 andboth components 1 and 2 respectively Finally the densityfunction for system 119894 at 119905

119894becomes sum

119895isin119904119894119888119894119875119894119895 Therefore

the applicable unified likelihood function for both hybridsystems and censoring schemes is

119871 (D) =

119903

prod

119894=1

sum

119895isin119904119894

119888119894119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

9prod

119896=1

119903119896

prod

119894=119903119896minus1+1

sum

119895isin119904119894

119875119894119895

[119875 (119879gt 120591)]

119899minus119903

= 119862

1199031

prod

119894=11198751198941

1199032

prod

119894=1199031+11198751198942

1199033

prod

119894=1199032+11198751198943

1199034

prod

119894=1199033+111987511989412

1199035

prod

119894=1199034+1(1198751198941 +1198751198942)

sdot

1199036

prod

119894=1199035+1(1198751198941 +1198751198943)

1199037

prod

119894=1199036+1(1198751198942 +1198751198943)

1199038

prod

119894=1199037+1(11987511989412 +119875

1198943)

sdot

119903

prod

119894=1199038+1(1198751198941 +1198751198942 +1198751198943) [119875 (119879gt 120591)]

119899minus119903

(10)

where the constant 119862 = prod

119903

119894=1119888119894 does not contain theparameters of interest in 119875

119894119895

For the purpose of simplicity we only consider twospecial cases of failure rates (1) the components were shockedby independent Poisson processes with same parameters thatis 1205821 = 1205822 = 12058212 = 120582 (2) the Poisson processes affectingthe three components individually have the same parametersbut different from that of the Poisson process applying oncomponents 1 and 2 simultaneously that is 1205821 = 1205822 = 120582 =

12058212 The maximum likelihood estimation (MLE) approachwill be implemented for the inference To make notationsimpler we denote the log-likelihood function as 119897(120579) =

log 119871(120579 | D) where 120579 is the parameter of failure rates includedin the life densities We also apply the approximated chi-squared likelihood ratio statistic [16] to numerically obtainthe confidence intervals of parameters Particularly for ourcase the likelihood ratio statistic for the parameter Λ =

minus2log[119871(120579)119871(120579)] approximately follows 120594

2] where 120579 = 120582 or

120579 = (120582 12058212) and its MLE 120579 and ] is the dimension of 120579

In general this method works well even for the situationof small sample size that is the coverage probability of theconstructed interval is very close to the nominal confidencelevel

31 Series-Parallel System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (6) and thedensities in (7) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(2120582119890minus3120582119905119894 minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=+1199033+1(120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(4120582119890minus3120582119905119894 minus 4120582119890minus4120582119905119894)

sdot

1199037

prod

119894=1199035+1(4120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199037+1(2120582119890minus3120582119905119894)

sdot

119903

prod

119894=1199038+1(6120582119890minus3120582119905119894 minus 5120582119890minus4120582119905119894)

times 119890

minus4(119899minus119903)120582120591(2119890120582120591 minus 1)

119899minus119903

(11)

So the log-likelihood can be simplified as

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591 + (119899 minus 119903) log (2119890120582120591 minus 1)

+

119903

sum

119894=1log 120582minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (2119890120582119905119894 minus 2)

+

1199033

sum

119894=1199032+1log (2119890120582119905119894 minus 1) +

1199035

sum

119894=1199034+1log (4119890120582119905119894 minus 4)

+

1199037

sum

119894=1199035+1log (4119890120582119905119894 minus 3) +

1199038

sum

119894=1199037+1log (2119890120582119905119894)

+

119903

sum

119894=1199038+1log (6119890120582119905119894 minus 5)

(12)

and its derivative with respect to 120582 is

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 +

2 (119899 minus 119903) 120591

2 minus 119890

minus120582120591+

119903

120582

minus

119903

sum

119894=14119905119894

+

1199032

sum

119894=1

119905119894

1 minus 119890

minus120582119905119894+

1199033

sum

119894=1199032+1(

2119905119894

2 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

119905119894

1 minus 119890

minus120582119905119894)+

1199037

sum

119894=1199035+1(

4119905119894

4 minus 3119890minus120582119905119894)

+

1199038

sum

119894=1199037+1119905119894+

119903

sum

119894=1199038+1(

6119905119894

6 minus 5119890minus120582119905119894)

(13)

Since no analytical form of MLE 120582 can be obtained from

the equation 119897

1015840(120582) = 0 a numerical method has to be

implemented for specific data observations The uniqueness

Advances in Statistics 5

of MLE can be justified in the following way the termsinvolving exponent in 119897

1015840(120582) can be expressed as a unified

functional form 119892(120582) = 119886(119887 minus 119888119890

minus120582119905) with positive constants

119886 119887 and 119888 Since 119892

1015840(120582) = minus119886119888119905119890

minus120582119905(119887 minus 119888119890

minus120582119905)

2lt 0 we have

119897

10158401015840(120582) = minus

2 (119899 minus 119903) 120591

2119890

minus120582120591

(2 minus 119890

minus120582120591)

2 minus

119903

120582

2 minus

1199032

sum

119894=1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199033

sum

119894=1199032+1

21199052119894119890

minus120582119905119894

(2 minus 119890

minus120582119905119894)

2 minus

1199035

sum

119894=1199034+1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199037

sum

119894=1199035+1

121199052119894119890

minus120582119905119894

(4 minus 3119890minus120582119905119894)2minus

119903

sum

119894=1199038+1

301199052119894119890

minus120582119905119894

(6 minus 5119890minus120582119905119894)2

lt 0

(14)

Hence the log-likelihood function 119897(120582) is strictly concave andtherefore 119897

1015840(120582) = 0 implies a unique MLE

120582 Additionallylim120582rarr 01198971015840(120582) = infin lim

120582rarrinfin119897

1015840(120582) = minus3(119899 minus 119903)120591 minus 3sum

119903

119894=1 119905119894lt 0

and so the MLE 120582 is a positive value

(2) 1205821

= 1205822

= 120582 = 12058212 Under this case the likelihood

function (10) reduces to

119871 (120582 12058212 | D)

= 119862

1199032

prod

119894=1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(3120582+12058212)119905119894

]

sdot

1199035

prod

119894=1199034+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

1199038

prod

119894=1199037+1[12058212119890minus(3120582+12058212)119905119894

+120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

119903

prod

119894=1199038+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)] 119890

minus(119899minus119903)(2120582+12058212)120591(2

minus 119890

minus120582120591)

119899minus119903

(15)

and so the log-likelihood function is

119897 (120582 12058212)

= log119862minus (119899 minus 119903) (2120582+12058212) 120591

+ (119899 minus 119903) log (2minus 119890

minus120582120591) minus

119903

sum

119894=1(2120582+12058212) 119905119894

+

1199032

sum

119894=1[log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199033

sum

119894=1199032+1[log 120582+ log (2minus 119890

minus120582119905119894)]

+

1199034

sum

119894=1199033+1(log 12058212 minus120582119905

119894)

+

1199035

sum

119894=1199034+1[log 2+ log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [(3120582+12058212) minus (2120582+12058212) 119890

minus120582119905119894]

+

1199038

sum

119894=1199037+1log [2120582minus (120582 minus 12058212) 119890

minus120582119905119894]

+

119903

sum

119894=1199038+1log [(4120582+ 212058212) minus (3120582+ 212058212) 119890

minus120582119905119894]

(16)

TheMLEs 120582 12058212 can be obtained numerically in the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The existence of MLE isprovided in the Appendix

32 Parallel-Series System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (8) and thedensities in (9) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(120582119890

minus120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=1199033+1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199035+1(120582119890

minus120582119905119894+120582119890

minus3120582119905119894minus 2120582119890minus4120582119905119894)

6 Advances in Statistics

sdot

119903

prod

119894=1199038+1(120582119890

minus120582119905119894+ 2120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

times (119890

minus120582120591+ 119890

minus3120582120591minus 119890

minus4120582120591)

119899minus119903

(17)

and then the log-likelihood function is

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591

+ (119899 minus 119903) log (119890

3120582120591+ 119890

120582120591minus 1) +

119903

sum

119894=1log 120582

minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (119890

120582119905119894minus 1)

+

1199033

sum

119894=1199032+1log (119890

3120582119905119894minus 1) +

1199034

sum

119894=1199033+1log (119890

120582119905119894minus 1)

+

1199035

sum

119894=1199034+1log (2119890120582119905119894 minus 2)

+

1199038

sum

119894=1199035+1log (119890

3120582119905119894+ 119890

120582119905119894minus 2)

+

119903

sum

119894=1199038+1log (119890

3120582119905119894+ 2119890120582119905119894 minus 3)

(18)

Taking derivative with respect to 120582 we obtain

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 + (119899 minus 119903)

3120591 + 120591119890

minus2120582120591

1 + 119890

minus2120582120591minus 119890

minus3120582120591 +119903

120582

minus

119903

sum

119894=14119905119894+

1199032

sum

119894=1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

3119905119894

1 minus 119890

minus3120582119905119894)+

1199035

sum

119894=1199033+1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199038

sum

119894=1199035+1(

3119905119894+ 119905119894119890

minus2120582119905119894

1 + 119890

minus2120582119905119894minus 2119890minus3120582119905119894

)

+

119903

sum

119894=1199038+1(

3119905119894+ 2119905119894119890

minus2120582119905119894

1 + 2119890minus2120582119905119894 minus 3119890minus3120582119905119894)

(19)

Since lim120582rarr 01198971015840(120582) = infin and lim

120582rarrinfin119897

1015840(120582) = minus(119899 minus 119903)120591 minus

sum

119903

119894=1 4119905119894 + sum

1199032119894=1 119905119894+ sum

1199033119894=1199032+1

3119905119894+ sum

1199035119894=1199033+1

119905119894+ sum

119903

119894=1199035+1 3119905119894 lt 0119897

1015840(120582) = 0 has a positive root 120582

(2) 1205821= 1205822= 120582 = 120582

12 Under this special case the likelihood

function (10) then reduces to

119871 (120582 12058212 | D) = 119862

1199032

prod

119894=1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)]

sdot

1199035

prod

119894=1199034+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199038

prod

119894=1199037+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

119903

prod

119894=1199038+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)] (119890

minus120582120591+ 119890

minus(2120582+12058212)120591

minus 119890

minus(3120582+12058212)120591)

119899minus119903

(20)

and so the log-likelihood function is

119897 (120582 12058212) = log119862minus (119899 minus 119903) 120582120591 + (119899 minus 119903) log [1

+ 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591] minus

119903

sum

119894=1120582119905119894+

1199032

sum

119894=1[log 120582minus (120582

+ 12058212) 119905119894 + log (1minus 119890

minus120582119905119894)] +

1199033

sum

119894=1199032+1[log 120582

+ log (1minus 119890

minus(2120582+12058212)119905119894)] +

1199034

sum

119894=1199033+1[log 12058212 minus (120582 + 12058212)

sdot 119905119894+ log (1minus 119890

minus120582119905119894)] +

1199035

sum

119894=1199034+1[log 2minus (120582 + 12058212) 119905119894

+ log 120582+ log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [120582119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

Advances in Statistics 7

Table 1 Series-parallel system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10481 00267 07164 9835 4 3 3 3 2 2 1 1 10 10214 00226 06926 9814 4 4 2 2 2 2 2 2 08 08308 00177 05664 9845 4 3 3 3 2 2 1 1 08 08297 00172 05652 988

30

5 5 5 3 3 3 2 2 2 10 10371 00226 06366 9726 6 5 3 3 2 2 2 2 10 10288 00212 06196 9625 5 5 3 3 3 2 2 2 08 08357 00153 05116 9786 6 5 3 3 2 2 2 1 08 08255 00142 04956 972

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

3 3 3 2 2 2 2 2 1 10 10370 00298 07645 9824 4 3 2 2 2 2 1 1 10 10323 00283 07290 9813 3 3 2 2 2 2 2 1 08 08307 00175 06117 9854 4 3 2 2 2 2 1 1 08 08264 00172 05879 994

30

5 4 5 3 2 2 2 2 1 10 10407 00271 06749 9745 5 4 3 3 2 2 2 2 10 10380 00245 06491 9835 4 5 3 2 2 2 2 1 08 08325 00157 05368 9755 5 4 3 3 2 2 2 2 08 08245 00157 05193 983

+120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

1199038

sum

119894=1199037+1log [12058212119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

119903

sum

119894=1199038+1log [2120582119890minus(120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

(21)

The MLEs 120582 12058212 will be obtained numerically in the equa-

tions 120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The prove of theexistence of MLE is given in the Appendix

4 Simulation Study

In this section we conduct a simulation study to investi-gate the performance of our methodology We choose twoparameter values of failure rates for each case in the twohybrid systems that is 120582 = 10 08 in the case of samefailure rates and (120582 12058212) = (10 05) (08 04) for the case ofdifferent failure rates Under each setting of parameter valueswe carry out simulation study to generate the lifetimes 119879

119894

119894 = 1 2 3 following the construction described in Section 21under two sample sizes 119899 = 24 30 for each of which twocomplete samples (119899 = 119903 = sum

9119896=1 119903119896) with two settings

of failure numbers 119903119896and two censored samples with two

failure numbers (119903 = 20 21 for 119899 = 24 and 119903 = 26 28for 119899 = 30) are considered to determine the sample size119899 and 119903

119896variation effects for the estimation precision We

conduct 10000 Monte-Carlo simulations for each setting

of parameter value sample size and failure number Theaveraged MLE mean squared error (MSE) length of 95confidence interval and coverage probability are displayed inTables 1 and 2 for the series-parallel system and Tables 3 and4 for the parallel-series system

In each table the estimation results in the upper panelcorrespond to the complete sample and lower panel to thecensored sample It seems that the estimations are reasonablygood under these relative small sample sizes and all thecoverage probabilities of confidence intervals exceed thenominal confidence level indicating that it is a conservativemethod for interval estimation by chi-squared likelihoodratio statistics As expected under the same sample size119899 the MSEs and interval lengths are smaller in completesamples than these in censored samples Due to the scale ofthe true parameter values we noticed that given the samesample size 119899 and failure numbers 119903

119896rsquos the MSE and interval

length of estimates under larger true parameter values areconsistently larger than these under smaller true values InTable 1 for example given 119899 = 24 119903 = 24 the MSE =00267 and 95 confidence interval length = 07164 when120582 = 1 whereas MSE = 00177 and the length = 05664when 120582 = 08 However it is common that for a faircomparison between estimates variability with different unitsor different parameter values one should use a relativevariability measure such as coefficient of variation instead ofa measure of dispersion like MSE or interval length In ourcase we propose a ldquonormalizedrdquo measure of dispersion 119877 =

lengthestimate to remove the scale effect for the comparisonAs a result the estimation results mentioned above give us119877 = 0716410481 = 06818 and 0566408308 = 06817respectively which are very close to each other Similaroutcomes are obtained for other estimation results across thetables indicating a consistent precision for the estimationprocedure

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 5

of MLE can be justified in the following way the termsinvolving exponent in 119897

1015840(120582) can be expressed as a unified

functional form 119892(120582) = 119886(119887 minus 119888119890

minus120582119905) with positive constants

119886 119887 and 119888 Since 119892

1015840(120582) = minus119886119888119905119890

minus120582119905(119887 minus 119888119890

minus120582119905)

2lt 0 we have

119897

10158401015840(120582) = minus

2 (119899 minus 119903) 120591

2119890

minus120582120591

(2 minus 119890

minus120582120591)

2 minus

119903

120582

2 minus

1199032

sum

119894=1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199033

sum

119894=1199032+1

21199052119894119890

minus120582119905119894

(2 minus 119890

minus120582119905119894)

2 minus

1199035

sum

119894=1199034+1

119905

2119894119890

minus120582119905119894

(1 minus 119890

minus120582119905119894)

2

minus

1199037

sum

119894=1199035+1

121199052119894119890

minus120582119905119894

(4 minus 3119890minus120582119905119894)2minus

119903

sum

119894=1199038+1

301199052119894119890

minus120582119905119894

(6 minus 5119890minus120582119905119894)2

lt 0

(14)

Hence the log-likelihood function 119897(120582) is strictly concave andtherefore 119897

1015840(120582) = 0 implies a unique MLE

120582 Additionallylim120582rarr 01198971015840(120582) = infin lim

120582rarrinfin119897

1015840(120582) = minus3(119899 minus 119903)120591 minus 3sum

119903

119894=1 119905119894lt 0

and so the MLE 120582 is a positive value

(2) 1205821

= 1205822

= 120582 = 12058212 Under this case the likelihood

function (10) reduces to

119871 (120582 12058212 | D)

= 119862

1199032

prod

119894=1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(3120582+12058212)119905119894

]

sdot

1199035

prod

119894=1199034+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[(120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

1199038

prod

119894=1199037+1[12058212119890minus(3120582+12058212)119905119894

+120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)]

sdot

119903

prod

119894=1199038+1[2 (120582 + 12058212) 119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus(2120582+12058212)119905119894(2minus 119890

minus120582119905119894)] 119890

minus(119899minus119903)(2120582+12058212)120591(2

minus 119890

minus120582120591)

119899minus119903

(15)

and so the log-likelihood function is

119897 (120582 12058212)

= log119862minus (119899 minus 119903) (2120582+12058212) 120591

+ (119899 minus 119903) log (2minus 119890

minus120582120591) minus

119903

sum

119894=1(2120582+12058212) 119905119894

+

1199032

sum

119894=1[log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199033

sum

119894=1199032+1[log 120582+ log (2minus 119890

minus120582119905119894)]

+

1199034

sum

119894=1199033+1(log 12058212 minus120582119905

119894)

+

1199035

sum

119894=1199034+1[log 2+ log (120582 + 12058212) + log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [(3120582+12058212) minus (2120582+12058212) 119890

minus120582119905119894]

+

1199038

sum

119894=1199037+1log [2120582minus (120582 minus 12058212) 119890

minus120582119905119894]

+

119903

sum

119894=1199038+1log [(4120582+ 212058212) minus (3120582+ 212058212) 119890

minus120582119905119894]

(16)

TheMLEs 120582 12058212 can be obtained numerically in the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The existence of MLE isprovided in the Appendix

32 Parallel-Series System

(1) 1205821= 1205822= 12058212

= 120582 Based on the reliability in (8) and thedensities in (9) the likelihood function (10) becomes

119871 (120582 | D) = 119862

1199032

prod

119894=1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199033

prod

119894=1199032+1(120582119890

minus120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199034

prod

119894=1199033+1(120582119890

minus3120582119905119894minus120582119890

minus4120582119905119894)

sdot

1199035

prod

119894=1199034+1(2120582119890minus3120582119905119894 minus 2120582119890minus4120582119905119894)

sdot

1199038

prod

119894=1199035+1(120582119890

minus120582119905119894+120582119890

minus3120582119905119894minus 2120582119890minus4120582119905119894)

6 Advances in Statistics

sdot

119903

prod

119894=1199038+1(120582119890

minus120582119905119894+ 2120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

times (119890

minus120582120591+ 119890

minus3120582120591minus 119890

minus4120582120591)

119899minus119903

(17)

and then the log-likelihood function is

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591

+ (119899 minus 119903) log (119890

3120582120591+ 119890

120582120591minus 1) +

119903

sum

119894=1log 120582

minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (119890

120582119905119894minus 1)

+

1199033

sum

119894=1199032+1log (119890

3120582119905119894minus 1) +

1199034

sum

119894=1199033+1log (119890

120582119905119894minus 1)

+

1199035

sum

119894=1199034+1log (2119890120582119905119894 minus 2)

+

1199038

sum

119894=1199035+1log (119890

3120582119905119894+ 119890

120582119905119894minus 2)

+

119903

sum

119894=1199038+1log (119890

3120582119905119894+ 2119890120582119905119894 minus 3)

(18)

Taking derivative with respect to 120582 we obtain

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 + (119899 minus 119903)

3120591 + 120591119890

minus2120582120591

1 + 119890

minus2120582120591minus 119890

minus3120582120591 +119903

120582

minus

119903

sum

119894=14119905119894+

1199032

sum

119894=1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

3119905119894

1 minus 119890

minus3120582119905119894)+

1199035

sum

119894=1199033+1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199038

sum

119894=1199035+1(

3119905119894+ 119905119894119890

minus2120582119905119894

1 + 119890

minus2120582119905119894minus 2119890minus3120582119905119894

)

+

119903

sum

119894=1199038+1(

3119905119894+ 2119905119894119890

minus2120582119905119894

1 + 2119890minus2120582119905119894 minus 3119890minus3120582119905119894)

(19)

Since lim120582rarr 01198971015840(120582) = infin and lim

120582rarrinfin119897

1015840(120582) = minus(119899 minus 119903)120591 minus

sum

119903

119894=1 4119905119894 + sum

1199032119894=1 119905119894+ sum

1199033119894=1199032+1

3119905119894+ sum

1199035119894=1199033+1

119905119894+ sum

119903

119894=1199035+1 3119905119894 lt 0119897

1015840(120582) = 0 has a positive root 120582

(2) 1205821= 1205822= 120582 = 120582

12 Under this special case the likelihood

function (10) then reduces to

119871 (120582 12058212 | D) = 119862

1199032

prod

119894=1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)]

sdot

1199035

prod

119894=1199034+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199038

prod

119894=1199037+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

119903

prod

119894=1199038+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)] (119890

minus120582120591+ 119890

minus(2120582+12058212)120591

minus 119890

minus(3120582+12058212)120591)

119899minus119903

(20)

and so the log-likelihood function is

119897 (120582 12058212) = log119862minus (119899 minus 119903) 120582120591 + (119899 minus 119903) log [1

+ 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591] minus

119903

sum

119894=1120582119905119894+

1199032

sum

119894=1[log 120582minus (120582

+ 12058212) 119905119894 + log (1minus 119890

minus120582119905119894)] +

1199033

sum

119894=1199032+1[log 120582

+ log (1minus 119890

minus(2120582+12058212)119905119894)] +

1199034

sum

119894=1199033+1[log 12058212 minus (120582 + 12058212)

sdot 119905119894+ log (1minus 119890

minus120582119905119894)] +

1199035

sum

119894=1199034+1[log 2minus (120582 + 12058212) 119905119894

+ log 120582+ log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [120582119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

Advances in Statistics 7

Table 1 Series-parallel system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10481 00267 07164 9835 4 3 3 3 2 2 1 1 10 10214 00226 06926 9814 4 4 2 2 2 2 2 2 08 08308 00177 05664 9845 4 3 3 3 2 2 1 1 08 08297 00172 05652 988

30

5 5 5 3 3 3 2 2 2 10 10371 00226 06366 9726 6 5 3 3 2 2 2 2 10 10288 00212 06196 9625 5 5 3 3 3 2 2 2 08 08357 00153 05116 9786 6 5 3 3 2 2 2 1 08 08255 00142 04956 972

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

3 3 3 2 2 2 2 2 1 10 10370 00298 07645 9824 4 3 2 2 2 2 1 1 10 10323 00283 07290 9813 3 3 2 2 2 2 2 1 08 08307 00175 06117 9854 4 3 2 2 2 2 1 1 08 08264 00172 05879 994

30

5 4 5 3 2 2 2 2 1 10 10407 00271 06749 9745 5 4 3 3 2 2 2 2 10 10380 00245 06491 9835 4 5 3 2 2 2 2 1 08 08325 00157 05368 9755 5 4 3 3 2 2 2 2 08 08245 00157 05193 983

+120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

1199038

sum

119894=1199037+1log [12058212119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

119903

sum

119894=1199038+1log [2120582119890minus(120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

(21)

The MLEs 120582 12058212 will be obtained numerically in the equa-

tions 120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The prove of theexistence of MLE is given in the Appendix

4 Simulation Study

In this section we conduct a simulation study to investi-gate the performance of our methodology We choose twoparameter values of failure rates for each case in the twohybrid systems that is 120582 = 10 08 in the case of samefailure rates and (120582 12058212) = (10 05) (08 04) for the case ofdifferent failure rates Under each setting of parameter valueswe carry out simulation study to generate the lifetimes 119879

119894

119894 = 1 2 3 following the construction described in Section 21under two sample sizes 119899 = 24 30 for each of which twocomplete samples (119899 = 119903 = sum

9119896=1 119903119896) with two settings

of failure numbers 119903119896and two censored samples with two

failure numbers (119903 = 20 21 for 119899 = 24 and 119903 = 26 28for 119899 = 30) are considered to determine the sample size119899 and 119903

119896variation effects for the estimation precision We

conduct 10000 Monte-Carlo simulations for each setting

of parameter value sample size and failure number Theaveraged MLE mean squared error (MSE) length of 95confidence interval and coverage probability are displayed inTables 1 and 2 for the series-parallel system and Tables 3 and4 for the parallel-series system

In each table the estimation results in the upper panelcorrespond to the complete sample and lower panel to thecensored sample It seems that the estimations are reasonablygood under these relative small sample sizes and all thecoverage probabilities of confidence intervals exceed thenominal confidence level indicating that it is a conservativemethod for interval estimation by chi-squared likelihoodratio statistics As expected under the same sample size119899 the MSEs and interval lengths are smaller in completesamples than these in censored samples Due to the scale ofthe true parameter values we noticed that given the samesample size 119899 and failure numbers 119903

119896rsquos the MSE and interval

length of estimates under larger true parameter values areconsistently larger than these under smaller true values InTable 1 for example given 119899 = 24 119903 = 24 the MSE =00267 and 95 confidence interval length = 07164 when120582 = 1 whereas MSE = 00177 and the length = 05664when 120582 = 08 However it is common that for a faircomparison between estimates variability with different unitsor different parameter values one should use a relativevariability measure such as coefficient of variation instead ofa measure of dispersion like MSE or interval length In ourcase we propose a ldquonormalizedrdquo measure of dispersion 119877 =

lengthestimate to remove the scale effect for the comparisonAs a result the estimation results mentioned above give us119877 = 0716410481 = 06818 and 0566408308 = 06817respectively which are very close to each other Similaroutcomes are obtained for other estimation results across thetables indicating a consistent precision for the estimationprocedure

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Statistics

sdot

119903

prod

119894=1199038+1(120582119890

minus120582119905119894+ 2120582119890minus3120582119905119894 minus 3120582119890minus4120582119905119894)

times (119890

minus120582120591+ 119890

minus3120582120591minus 119890

minus4120582120591)

119899minus119903

(17)

and then the log-likelihood function is

119897 (120582) = log119862minus 4 (119899 minus 119903) 120582120591

+ (119899 minus 119903) log (119890

3120582120591+ 119890

120582120591minus 1) +

119903

sum

119894=1log 120582

minus

119903

sum

119894=14120582119905119894+

1199032

sum

119894=1log (119890

120582119905119894minus 1)

+

1199033

sum

119894=1199032+1log (119890

3120582119905119894minus 1) +

1199034

sum

119894=1199033+1log (119890

120582119905119894minus 1)

+

1199035

sum

119894=1199034+1log (2119890120582119905119894 minus 2)

+

1199038

sum

119894=1199035+1log (119890

3120582119905119894+ 119890

120582119905119894minus 2)

+

119903

sum

119894=1199038+1log (119890

3120582119905119894+ 2119890120582119905119894 minus 3)

(18)

Taking derivative with respect to 120582 we obtain

119897

1015840(120582) = minus 4 (119899 minus 119903) 120591 + (119899 minus 119903)

3120591 + 120591119890

minus2120582120591

1 + 119890

minus2120582120591minus 119890

minus3120582120591 +119903

120582

minus

119903

sum

119894=14119905119894+

1199032

sum

119894=1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

3119905119894

1 minus 119890

minus3120582119905119894)+

1199035

sum

119894=1199033+1(

119905119894

1 minus 119890

minus120582119905119894)

+

1199038

sum

119894=1199035+1(

3119905119894+ 119905119894119890

minus2120582119905119894

1 + 119890

minus2120582119905119894minus 2119890minus3120582119905119894

)

+

119903

sum

119894=1199038+1(

3119905119894+ 2119905119894119890

minus2120582119905119894

1 + 2119890minus2120582119905119894 minus 3119890minus3120582119905119894)

(19)

Since lim120582rarr 01198971015840(120582) = infin and lim

120582rarrinfin119897

1015840(120582) = minus(119899 minus 119903)120591 minus

sum

119903

119894=1 4119905119894 + sum

1199032119894=1 119905119894+ sum

1199033119894=1199032+1

3119905119894+ sum

1199035119894=1199033+1

119905119894+ sum

119903

119894=1199035+1 3119905119894 lt 0119897

1015840(120582) = 0 has a positive root 120582

(2) 1205821= 1205822= 120582 = 120582

12 Under this special case the likelihood

function (10) then reduces to

119871 (120582 12058212 | D) = 119862

1199032

prod

119894=1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)]

sdot

1199033

prod

119894=1199032+1[120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199034

prod

119894=1199033+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)]

sdot

1199035

prod

119894=1199034+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)]

sdot

1199037

prod

119894=1199035+1[120582119890

minus(2120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

1199038

prod

119894=1199037+1[12058212119890minus(2120582+12058212)119905119894

(1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)]

sdot

119903

prod

119894=1199038+1[2120582119890minus(2120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582119890

minus120582119905119894(1minus 119890

minus(2120582+12058212)119905119894)] (119890

minus120582120591+ 119890

minus(2120582+12058212)120591

minus 119890

minus(3120582+12058212)120591)

119899minus119903

(20)

and so the log-likelihood function is

119897 (120582 12058212) = log119862minus (119899 minus 119903) 120582120591 + (119899 minus 119903) log [1

+ 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591] minus

119903

sum

119894=1120582119905119894+

1199032

sum

119894=1[log 120582minus (120582

+ 12058212) 119905119894 + log (1minus 119890

minus120582119905119894)] +

1199033

sum

119894=1199032+1[log 120582

+ log (1minus 119890

minus(2120582+12058212)119905119894)] +

1199034

sum

119894=1199033+1[log 12058212 minus (120582 + 12058212)

sdot 119905119894+ log (1minus 119890

minus120582119905119894)] +

1199035

sum

119894=1199034+1[log 2minus (120582 + 12058212) 119905119894

+ log 120582+ log (1minus 119890

minus120582119905119894)]

+

1199037

sum

119894=1199035+1log [120582119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

Advances in Statistics 7

Table 1 Series-parallel system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10481 00267 07164 9835 4 3 3 3 2 2 1 1 10 10214 00226 06926 9814 4 4 2 2 2 2 2 2 08 08308 00177 05664 9845 4 3 3 3 2 2 1 1 08 08297 00172 05652 988

30

5 5 5 3 3 3 2 2 2 10 10371 00226 06366 9726 6 5 3 3 2 2 2 2 10 10288 00212 06196 9625 5 5 3 3 3 2 2 2 08 08357 00153 05116 9786 6 5 3 3 2 2 2 1 08 08255 00142 04956 972

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

3 3 3 2 2 2 2 2 1 10 10370 00298 07645 9824 4 3 2 2 2 2 1 1 10 10323 00283 07290 9813 3 3 2 2 2 2 2 1 08 08307 00175 06117 9854 4 3 2 2 2 2 1 1 08 08264 00172 05879 994

30

5 4 5 3 2 2 2 2 1 10 10407 00271 06749 9745 5 4 3 3 2 2 2 2 10 10380 00245 06491 9835 4 5 3 2 2 2 2 1 08 08325 00157 05368 9755 5 4 3 3 2 2 2 2 08 08245 00157 05193 983

+120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

1199038

sum

119894=1199037+1log [12058212119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

119903

sum

119894=1199038+1log [2120582119890minus(120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

(21)

The MLEs 120582 12058212 will be obtained numerically in the equa-

tions 120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The prove of theexistence of MLE is given in the Appendix

4 Simulation Study

In this section we conduct a simulation study to investi-gate the performance of our methodology We choose twoparameter values of failure rates for each case in the twohybrid systems that is 120582 = 10 08 in the case of samefailure rates and (120582 12058212) = (10 05) (08 04) for the case ofdifferent failure rates Under each setting of parameter valueswe carry out simulation study to generate the lifetimes 119879

119894

119894 = 1 2 3 following the construction described in Section 21under two sample sizes 119899 = 24 30 for each of which twocomplete samples (119899 = 119903 = sum

9119896=1 119903119896) with two settings

of failure numbers 119903119896and two censored samples with two

failure numbers (119903 = 20 21 for 119899 = 24 and 119903 = 26 28for 119899 = 30) are considered to determine the sample size119899 and 119903

119896variation effects for the estimation precision We

conduct 10000 Monte-Carlo simulations for each setting

of parameter value sample size and failure number Theaveraged MLE mean squared error (MSE) length of 95confidence interval and coverage probability are displayed inTables 1 and 2 for the series-parallel system and Tables 3 and4 for the parallel-series system

In each table the estimation results in the upper panelcorrespond to the complete sample and lower panel to thecensored sample It seems that the estimations are reasonablygood under these relative small sample sizes and all thecoverage probabilities of confidence intervals exceed thenominal confidence level indicating that it is a conservativemethod for interval estimation by chi-squared likelihoodratio statistics As expected under the same sample size119899 the MSEs and interval lengths are smaller in completesamples than these in censored samples Due to the scale ofthe true parameter values we noticed that given the samesample size 119899 and failure numbers 119903

119896rsquos the MSE and interval

length of estimates under larger true parameter values areconsistently larger than these under smaller true values InTable 1 for example given 119899 = 24 119903 = 24 the MSE =00267 and 95 confidence interval length = 07164 when120582 = 1 whereas MSE = 00177 and the length = 05664when 120582 = 08 However it is common that for a faircomparison between estimates variability with different unitsor different parameter values one should use a relativevariability measure such as coefficient of variation instead ofa measure of dispersion like MSE or interval length In ourcase we propose a ldquonormalizedrdquo measure of dispersion 119877 =

lengthestimate to remove the scale effect for the comparisonAs a result the estimation results mentioned above give us119877 = 0716410481 = 06818 and 0566408308 = 06817respectively which are very close to each other Similaroutcomes are obtained for other estimation results across thetables indicating a consistent precision for the estimationprocedure

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 7

Table 1 Series-parallel system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10481 00267 07164 9835 4 3 3 3 2 2 1 1 10 10214 00226 06926 9814 4 4 2 2 2 2 2 2 08 08308 00177 05664 9845 4 3 3 3 2 2 1 1 08 08297 00172 05652 988

30

5 5 5 3 3 3 2 2 2 10 10371 00226 06366 9726 6 5 3 3 2 2 2 2 10 10288 00212 06196 9625 5 5 3 3 3 2 2 2 08 08357 00153 05116 9786 6 5 3 3 2 2 2 1 08 08255 00142 04956 972

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE(120582) 95 CI length CP ()

24

3 3 3 2 2 2 2 2 1 10 10370 00298 07645 9824 4 3 2 2 2 2 1 1 10 10323 00283 07290 9813 3 3 2 2 2 2 2 1 08 08307 00175 06117 9854 4 3 2 2 2 2 1 1 08 08264 00172 05879 994

30

5 4 5 3 2 2 2 2 1 10 10407 00271 06749 9745 5 4 3 3 2 2 2 2 10 10380 00245 06491 9835 4 5 3 2 2 2 2 1 08 08325 00157 05368 9755 5 4 3 3 2 2 2 2 08 08245 00157 05193 983

+120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

1199038

sum

119894=1199037+1log [12058212119890

minus(120582+12058212)119905119894(1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

+

119903

sum

119894=1199038+1log [2120582119890minus(120582+12058212)119905119894 (1minus 119890

minus120582119905119894)

+ 120582 (1minus 119890

minus(2120582+12058212)119905119894)]

(21)

The MLEs 120582 12058212 will be obtained numerically in the equa-

tions 120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0 The prove of theexistence of MLE is given in the Appendix

4 Simulation Study

In this section we conduct a simulation study to investi-gate the performance of our methodology We choose twoparameter values of failure rates for each case in the twohybrid systems that is 120582 = 10 08 in the case of samefailure rates and (120582 12058212) = (10 05) (08 04) for the case ofdifferent failure rates Under each setting of parameter valueswe carry out simulation study to generate the lifetimes 119879

119894

119894 = 1 2 3 following the construction described in Section 21under two sample sizes 119899 = 24 30 for each of which twocomplete samples (119899 = 119903 = sum

9119896=1 119903119896) with two settings

of failure numbers 119903119896and two censored samples with two

failure numbers (119903 = 20 21 for 119899 = 24 and 119903 = 26 28for 119899 = 30) are considered to determine the sample size119899 and 119903

119896variation effects for the estimation precision We

conduct 10000 Monte-Carlo simulations for each setting

of parameter value sample size and failure number Theaveraged MLE mean squared error (MSE) length of 95confidence interval and coverage probability are displayed inTables 1 and 2 for the series-parallel system and Tables 3 and4 for the parallel-series system

In each table the estimation results in the upper panelcorrespond to the complete sample and lower panel to thecensored sample It seems that the estimations are reasonablygood under these relative small sample sizes and all thecoverage probabilities of confidence intervals exceed thenominal confidence level indicating that it is a conservativemethod for interval estimation by chi-squared likelihoodratio statistics As expected under the same sample size119899 the MSEs and interval lengths are smaller in completesamples than these in censored samples Due to the scale ofthe true parameter values we noticed that given the samesample size 119899 and failure numbers 119903

119896rsquos the MSE and interval

length of estimates under larger true parameter values areconsistently larger than these under smaller true values InTable 1 for example given 119899 = 24 119903 = 24 the MSE =00267 and 95 confidence interval length = 07164 when120582 = 1 whereas MSE = 00177 and the length = 05664when 120582 = 08 However it is common that for a faircomparison between estimates variability with different unitsor different parameter values one should use a relativevariability measure such as coefficient of variation instead ofa measure of dispersion like MSE or interval length In ourcase we propose a ldquonormalizedrdquo measure of dispersion 119877 =

lengthestimate to remove the scale effect for the comparisonAs a result the estimation results mentioned above give us119877 = 0716410481 = 06818 and 0566408308 = 06817respectively which are very close to each other Similaroutcomes are obtained for other estimation results across thetables indicating a consistent precision for the estimationprocedure

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Statistics

Table 2 Series-parallel system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 4 3 3 2 2 2 2 3 10 05 09912 05285 00247 00168 08265 05748 9744 4 4 2 2 2 2 2 2 10 05 10434 04604 00238 00161 08236 05597 9853 4 3 3 2 2 2 2 3 08 04 07880 04231 00166 00153 06673 04716 9724 4 4 2 2 2 2 2 2 08 04 08347 03685 00155 00145 06654 04668 984

30

4 4 4 3 3 3 3 3 3 10 05 10328 05270 00189 00062 07190 05542 9765 5 5 3 3 3 2 2 2 10 05 10161 05317 00187 00058 07164 05502 9854 4 4 3 3 3 3 3 3 08 04 08362 04276 00172 00055 05835 04487 9805 5 5 3 3 3 2 2 2 08 04 08273 04344 00151 00052 05762 04452 987

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

3 3 2 2 2 2 2 2 2 10 05 10196 04803 00260 00188 08344 05909 9823 3 3 2 2 2 2 2 2 10 05 10388 04554 00257 00182 08288 05792 9763 3 2 2 2 2 2 2 2 08 04 08446 03985 00304 00162 06913 04898 9643 3 3 2 2 2 2 2 2 08 04 08621 03788 00279 00161 06882 04730 956

30

4 4 4 3 3 2 2 2 2 10 05 10180 05496 00202 00087 07425 05616 9765 4 4 3 3 3 2 2 2 10 05 10247 05476 00196 00082 07216 05545 9724 4 4 3 3 2 2 2 2 08 04 08300 04477 00188 00079 06053 04529 9765 4 4 3 3 3 2 2 2 08 04 08302 04432 00170 00067 05845 04490 980

Table 3 Parallel-series system 1205821 = 1205822 = 12058212 = 120582 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 4 4 2 2 2 2 2 2 10 10181 00257 06311 9645 4 3 3 3 2 2 1 1 10 10062 00219 06092 9664 4 4 2 2 2 2 2 2 08 08115 00149 05004 9645 4 3 2 2 2 2 1 1 08 08045 00138 04865 966

30

5 4 5 3 3 3 2 2 3 10 10049 00194 05546 9575 5 5 3 3 3 2 2 2 10 10022 00185 05543 9615 4 5 3 3 3 2 2 3 08 08038 00119 04412 9575 5 5 3 3 3 2 2 2 08 08002 00118 04406 961

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582

120582 MSE 95 CI length CP ()

24

4 3 2 2 2 2 2 1 2 10 10050 00266 06681 9693 3 3 2 2 2 2 2 2 10 10087 00259 06619 9574 3 2 2 2 2 2 1 2 08 08085 00166 05369 9783 3 3 2 2 2 2 2 2 08 08093 00163 05313 965

30

4 4 3 3 3 3 2 2 2 10 10065 00210 05874 9724 4 4 4 3 3 2 2 2 10 10102 00204 05733 9624 4 3 3 3 3 2 2 2 08 08052 00130 04700 9724 4 4 4 3 3 2 2 2 08 07991 00126 04514 951

Additionally other findings can be seen from the estima-tion results (i) For the complete samples the upper panelsin the tables interestingly show that given the same size 119899the MSEs and interval lengths are consistently smaller in thesetting of larger variation of 119903

119896rsquos than those in the setting

of less variation of 119903119896rsquos In other words the estimations are

more efficient under ldquounbalancedrdquo failure numbers (119903119896rsquos vary

largely) than ldquobalancedrdquo failure numbers (119903119896rsquos are close to each

other) The possible reason is that the likelihood functionwith ldquounbalancedrdquo failure numbers is less dispersed so that itaccommodates more amount of information of parameters(ii) For the censored samples the MSE and interval lengthare getting smaller as the sample size 119899 and failure number 119903

are getting larger For example for the true parameter values(120582 12058212) = (10 05) in the lower panel of Table 2 when 119899 = 30119903 = 28 the MSE(120582 12058212) = (00196 00082) and the intervallengths for 120582 12058212 07216 and 05545 respectively while thecorresponding MSE(120582 12058212) = (00257 00182) and intervallengths for 120582 12058212 08288 and 05792 under 119899 = 24 119903 = 21Furthermore given the sample size 119899 = 24 the MSE andinterval length under 119903 = 21 are smaller than these under119899 = 20 where the MSE(120582 12058212) = (00260 00188) and theinterval lengths of 120582 12058212 08344 and 05909 In summary theresults indicate that it is more accurate for the estimates ifmore failures are observed

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 9

Table 4 Parallel-series system 1205821 = 1205822 = 120582 = 12058212 (CP = coverage probability)

119899 (= 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 3 3 3 3 3 3 10 05 10082 04556 00214 00142 06364 05473 9843 3 3 3 2 2 2 2 4 10 05 10319 04560 00196 00113 06350 05439 9822 2 2 3 3 3 3 3 3 08 04 08258 03637 00173 00104 05097 04172 9703 3 3 3 2 2 2 2 4 08 04 08227 03625 00181 00113 05081 04162 976

30

2 2 3 3 4 4 4 4 4 10 05 09921 04522 00191 00191 05488 04903 9783 2 2 3 4 4 4 4 4 10 05 09880 04602 00152 00121 05474 04811 9842 2 3 3 4 4 4 4 4 08 04 08150 03691 00125 00107 04498 04295 9823 2 2 3 4 4 4 4 4 08 04 07900 03710 00125 00077 04488 04261 980

119899 (gt 119903) 119903119896 119896 = 1 2 9 120582 12058212

120582

12058212 MSE(120582 12058212) 95 CI length (120582 12058212) CP ()

24

2 2 2 2 2 2 2 3 3 10 05 09731 04589 00236 00180 06981 06887 9762 2 2 2 2 2 3 3 3 10 05 09890 04624 00172 00151 06628 06728 9722 2 2 2 2 2 2 3 3 08 04 08265 03638 00187 00086 05837 05771 9862 2 2 2 2 2 3 3 3 08 04 08301 03635 00179 00069 05644 05655 984

30

2 2 3 3 3 3 3 3 4 10 05 09551 04645 00169 00049 05982 06193 9662 2 3 3 3 3 4 4 4 10 05 09587 04627 00119 00039 05852 06047 9742 2 3 3 3 3 3 3 4 08 04 07884 03882 00100 00031 04937 05152 9802 2 3 3 3 3 4 4 4 08 04 07967 03869 00086 00031 04865 05087 978

5 Conclusions and Discussions

In this paper we have studied statistical inference for three-component hybrid systems based on masked data for whichthe lifetimes of units are nonindependent and nonidenticaldistributed Two commonly censored schemes type-I andtype-II were considered in the analysis We have presentedthe maximum likelihood estimates of parameters when thefailure rates of three components in the hybrid systemwere assumed to be the same and different respectivelyIn addition we obtained the approximate interval estima-tion of parameters by using likelihood ratio statistic Wehave assessed the performance of estimation methods bysimulation studies The results have demonstrated that theprocedure can achieve good estimation performances undersmall and moderate sample sizes and the estimates aremore accurate if more failures are observed indicating theefficiency of the estimation method While the methodcan be extended to more complex systems in the presenceof masked data the representation and evaluation of thelikelihood function would become cumbersome for largesystems There is an alternative method based on signaturethat explores component topology The system signature isthe probability vector whose element is the probability ofeach component failure resulting in the system failure and itprovides an elegantly simple representation of a system [17]Some advances and various applications of the signature arediscussed in [18ndash20] Recently using the system signaturea Bayesian inference to the system with masked lifetimedata was proposed by Aslett [21] The generic likelihoodfunction for complex systems can be easily expressed by dataaugmentation method the parameter inference is relied onthe samples from an iterative Markov chain Monte-Carlosimulation of all the component failure times and parametersThis intensive computing method provides an alternative

to the traditional likelihood-based approach to deal withgeneral systems

Appendices

Proof of existence of MLEs for the likelihood function underthe case 1205821 = 1205822 = 120582 = 12058212 in both hybrid systems

A Series-Parallel System

In the log-likelihood function in (16) taking partial deriva-tives with respect to 120582 and 12058212 respectively

120597119897 (120582 12058212)

120597120582

= minus 2 (119899 minus 119903) 120591 + (119899 minus 119903)

120591119890

minus120582120591

2 minus 119890

minus120582120591minus

119903

sum

119894=12119905119894

+

1199032

sum

119894=1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

119905119894119890

minus120582119905119894

2 minus 119890

minus120582119905119894)minus

1199034

sum

119894=1199033+1119905119894

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199037

sum

119894=1199035+1(

3 + [(2120582 + 12058212) 119905119894 minus 2] 119890minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Statistics

+

1199038

sum

119894=1199037+1(

2 + [(120582 minus 12058212) 119905119894 minus 1] 119890minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

4 + [(3120582 + 212058212) 119905119894 minus 3] 119890minus120582119905119894

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

120597119897 (120582 12058212)

12059712058212

= minus (119899 minus 119903) 120591 minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582 + 12058212

)+

1199034

sum

119894=1199033+1

112058212

+

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

+

1199037

sum

119894=1199035+1(

1 minus 119890

minus120582119905119894

(3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

+

1199038

sum

119894=1199037+1(

119890

minus120582119905119894

2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

+

119903

sum

119894=1199038+1(

2 (1 minus 119890

minus120582119905119894)

(4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

(A1)

First we notice that(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 it is easily seen

that lim120582rarr 01198921(120582) = infin and lim

120582rarrinfin1198921(120582) = minus2(119899 minus

119903)120591minussum

119903

119894=1 2119905119894minussum

1199034119894=1199033+1

119905119894lt 0 so there is a positive root

120582 for 1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we have

119892

1015840

2 (12058212)

= minus

1199032

sum

119894=1(

1120582 + 12058212

)

2minus

1199034

sum

119894=1199033+1(

112058212

)

2

minus

1199035

sum

119894=1199034+1(

1120582 + 12058212

)

2

minus

1199037

sum

119894=1199035+1

(1 minus 119890

minus120582119905119894)

2

((3120582 + 12058212) minus (2120582 + 12058212) 119890minus120582119905119894

)

2

minus

1199038

sum

119894=1199037+1

119890

minus2120582119905119894

(2120582 minus (120582 minus 12058212) 119890minus120582119905119894

)

2

minus

119903

sum

119894=1199038+1

4 (1 minus 119890

minus120582119905119894)

2

((4120582 + 212058212) minus (3120582 + 212058212) 119890minus120582119905119894

)

2 lt 0

(A2)

so 1198922(12058212) is decreasing for 12058212 Additionallylim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minus(119899 minus 119903)120591 minus sum

119903

119894=1 119905119894

lt 0 Thus 1198922(12058212) = 0 has aunique positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

B Parallel-Series System

For the log-likelihood function in (21) the partial derivativeswith respect to 120582 and 12058212 are

120597119897 (120582 12058212)

120597120582

= minus (119899 minus 119903) 120591 + (119899 minus 119903) (

minus120591119890

minus(120582+12058212)120591+ 2120591119890minus(2120582+12058212)120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)

minus

119903

sum

119894=1119905119894+

1199032

sum

119894=1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199033

sum

119894=1199032+1(

1120582

+

2119905119894119890

minus(2120582+12058212)119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(minus119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

minus120582119905119894)

+

1199035

sum

119894=1199034+1(

1120582

minus 119905119894+

119905119894119890

minus120582119905119894

1 minus 119890

120582119905119894)

+

1199037

sum

119894=1199035+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(1 minus 4119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

1 minus 12058212119905119894119890minus(120582+12058212)119905119894

+ [2 (120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(

1120582

minus

119905119894119890

minus(120582+12058212)119905119894(2 minus 6119890minus120582119905119894)

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

120597119897 (120582 12058212)

12059712058212= (119899 minus 119903) (minus120591 +

120591

1 + 119890

minus(120582+12058212)120591minus 119890

minus(2120582+12058212)120591)minus

1199032

sum

119894=1119905119894

+

1199033

sum

119894=1199032+1(minus119905119894+

119905119894

1 minus 119890

minus(2120582+12058212)119905119894)+

1199034

sum

119894=1199033+1(

112058212

minus 119905119894)minus

1199035

sum

119894=1199034+1119905119894

+

1199037

sum

119894=1199035+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(1 minus 2119890minus120582119905119894)

)

+

1199038

sum

119894=1199037+1(

(1 minus 12058212119905119894) 119890minus(120582+12058212)119905119894

+ [(120582 + 12058212) 119905119894 minus 1] 119890minus(2120582+12058212)119905119894

120582 + 12058212119890minus(120582+12058212)119905119894

minus (120582 + 12058212) 119890minus(2120582+12058212)119905119894

)

+

119903

sum

119894=1199038+1(minus119905119894+

119905119894

1 + 119890

minus(120582+12058212)119905119894(2 minus 3119890minus120582119905119894)

)

(B1)

It is worth noting that

(i) given 12058212 let 1198921(120582) = 120597119897(120582 12058212)120597120582 and lim120582rarr 01198921(120582)

= infin and lim120582rarrinfin

1198921(120582) = minus(119899minus119903)120591minussum

119903

119894=1 119905119894minussum

1199032119894=1 119905119894minus

sum

1199035119894=1199033+1

119905119894lt 0 so there is a positive root 120582 of1198921(120582) = 0

(ii) given 120582 let 1198922(12058212) = 120597119897(120582 12058212)12059712058212 we havelim12058212rarr 01198922(12058212) = infin and lim

12058212rarrinfin1198922(12058212) =

minussum

1199032119894=1 119905119894minus sum

1199035119894=1199033+1

119905119894

lt 0 Thus 1198922(12058212) = 0 hasa positive root

12058212 Hence the MLEs 120582

12058212 existand can be obtained numerically from the equations120597119897(120582 12058212)120597120582 = 120597119897(120582 12058212)12059712058212 = 0

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 11

Conflict of Interests

The authors declarethat there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Sharsquos work was partially supported by NSF CMMI-0654417and NIH NIMHD-2G12MD007592 Wangrsquos work was sup-ported by the National Statistical Scientific Research (Plan)Project (2013LZ08) Xursquos work was supported by the grantfrom Innovation Program of Shanghai Municipal EducationCommission (14YZ080 14ZZ155)

References

[1] J S Usher and T J Hodgson ldquoMaximum likelihood analysis ofcomponent reliability using masked system life-test datardquo IEEETransactions on Reliability vol 37 no 5 pp 550ndash555 1988

[2] NDoganaksoy ldquoInterval estimation fromcensored andmaskedsystem-failure datardquo IEEE Transactions on Reliability vol 40no 3 pp 280ndash286 1991

[3] D K J Lin J S Usher and FM Guess ldquoExact maximum likeli-hood estimation using masked system datardquo IEEE Transactionson Reliability vol 42 no 4 pp 631ndash635 1993

[4] B Reiser I Guttman D K Lin F M Guess and J S UsherldquoBayesian inference for masked system lifetime datardquo Journal ofthe Royal Statistical Society Series C Applied Statistics vol 44no 1 pp 79ndash90 1995

[5] D K J Lin J S Usher and F M Guess ldquoBayes estimationof component-reliability from masked system-life Datardquo IEEETransactions on Reliability vol 45 no 2 pp 233ndash237 1996

[6] J S Usher ldquoWeibull component reliability-prediction in thepresence of masked datardquo IEEE Transactions on Reliability vol45 no 2 pp 229ndash232 1996

[7] A M Sarhan and A H El-Bassiouny ldquoEstimation of compo-nents reliability in a parallel system using masked system lifedatardquo Applied Mathematics and Computation vol 138 no 1 pp61ndash75 2003

[8] A M Sarhan ldquoThe Bayes procedure in exponential reliabilityfamily models using conjugate convex tent prior familyrdquo Reli-ability Engineering and System Safety vol 71 no 1 pp 97ndash1022001

[9] A M Sarhan ldquoParameter estimations in linear failure ratemodel using masked datardquo Applied Mathematics and Compu-tation vol 151 no 1 pp 233ndash249 2004

[10] A M Sarhan and D Kundu ldquoBayes estimators for reliabilitymeasures in geometric distributionmodel usingmasked systemlife test datardquo Computational Statistics and Data Analysis vol52 no 4 pp 1821ndash1836 2008

[11] H Y Jiang and G F Zhang ldquoParameter estimation in exponen-tial failure rate model using masked datardquo Journal of ZhejiangUniversity vol 33 no 2 pp 125ndash128 2006

[12] A I El-Gohary ldquoBayesian estimation of the parameters intwonon-independent component series systemwith dependenttime failure raterdquo Applied Mathematics and Computation vol154 no 1 pp 41ndash51 2004

[13] F Zhang and Y Shi ldquoParameter estimation of the aerospacepower supply system using masked lifetime datardquo AerospaceControl vol 27 no 4 pp 96ndash100 2009

[14] Y Liu and Y Shi ldquoStatistical analysis of the reliability forpower supply of spacecraft with masked system life test datardquoAerospace Control vol 28 no 2 pp 70ndash74 2010

[15] A W Marshall and I Olkin ldquoA multivariate exponentialdistributionrdquo Journal of the American Statistical Association vol62 pp 30ndash44 1967

[16] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 1982

[17] F J Samaniego System Signatures and their Applications in Engi-neering Reliability vol 110 of International Series in OperationsResearch amp Management Science Springer Berlin Germany2007

[18] S Eryilmaz ldquoReview of recent advances in reliability of con-secutive k-out-of-n and related systemsrdquo Proceedings of theInstitution of Mechanical Engineers Part O Journal of Risk andReliability vol 224 no 3 pp 225ndash237 2010

[19] N Balakrishnan H K Ng and J Navarro ldquoExact nonpara-metric inference for component lifetime distribution based onlifetime data from systems with known signaturesrdquo Journal ofNonparametric Statistics vol 23 no 3 pp 741ndash752 2011

[20] F P A Coolen and T Coolen-Maturi ldquoGeneralizing the signa-ture to systems with multiple types of componentsrdquo in ComplexSystems and Dependability W Zamojski J Mazurkiewicz JSugier TWalkowiak and J Kacprzyk Eds vol 170 ofAdvancesin Intelligent and Soft Computing pp 115ndash130 Springer BerlinGermany 2012

[21] L J M Aslett MCMC for inference on phase-type and maskedsystem lifetime models [PhD thesis] Trinity College DublinDublin Ireland 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of