Research ArticleA New Lifetime Distribution and Its Power Transformation

Ammar M Sarhan1 Lotfi Tadj2 and David C Hamilton1

1 Department of Mathematics Dalhousie University Halifax NS Canada B3H 4R22Department of Information Systems and Decision Sciences Fairleigh Dickinson University Vancouver BC Canada V6B 2P6

Correspondence should be addressed to Ammar M Sarhan asarhan0yahoocom

Received 11 January 2014 Revised 8 April 2014 Accepted 12 April 2014 Published 18 May 2014

Academic Editor Chin-Shang Li

Copyright copy 2014 Ammar M Sarhan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Newone-parameter and two-parameter distributions are introduced in this paperThe failure rate of the one-parameter distributionis unimodal (upside-down bathtub) while the failure rate of the two-parameter distribution can be decreasing increasingunimodal increasing-decreasing-increasing or decreasing-increasing-decreasing depending on the values of its two parametersThe two-parameter distribution is derived from the one-parameter distribution by using a power transformation We discuss someproperties of these two distributions such as the behavior of the failure rate function the probability density function themomentsskewness and kurtosis and limiting distributions of order statistics Maximum likelihood estimation for the two-parameter modelusing complete samples is investigated Different algorithms for generating random samples from the two new models are givenApplications to real data are discussed and compared with the fit attained by some one- and two-parameter distributions Finally asimulation study is carried out to investigate themean square error of themaximum likelihood estimators the coverage probabilityand the width of the confidence intervals of the unknown parameters

1 Introduction

Lindley [1] proposed a one-parameter distribution nowknown as the Lindley distribution with the following proba-bility density function (pdf)

119891 (119905) =1205732

120573 + 1(1 + 119905) 119890

minus120573119905 119905 gt 0 120573 gt 0 (1)

The failure rate function of the Lindley distribution is alwaysincreasing The properties of the Lindley distribution arestudied in detail by Ghitany et al [2] There are situationsin which the Lindley distribution may not be suitable froma theoretical or applied point of view Ghitany et al [3] Forthis reason Ghitany et al [3] used a power transformation119883 = 1198791120572 to introduce the power Lindley distribution whichis a more flexible distribution The pdf of PL(120572 120573) is

119891 (119909) =1205721205732

120573 + 1(1 + 119909120572) 119890minus120573119909

120572

119909 gt 0 120572 120573 gt 0 (2)

Ghitany et al [3] showed that the hazard function of PL(120572 120573)can be increasing decreasing and decreasing-increasing-decreasing depending on the values of the parameters Theyalso discussed some of the statistical properties of the distri-bution and used themaximum likelihoodmethod to estimateits two unknown parameters and applied it to a real data setIn spite of the flexibility of the PL(120572 120573) to fit some real datasets it fails to fit some other data sets

The main aim of this paper is to introduce two newdistributions The first is a one-parameter distribution whichis similar to the Lindley distribution and the second isthe power transformation of the one-parameter distribu-tion We refer to these two distributions as 119873(120573) andTN(120572 120573) respectively The hazard function of 119873(120573) is onlyunimodal while the hazard function of TN(120572 120573) can bedecreasing increasing unimodal decreasing-increasing-decreasing or increasing-decreasing-increasing dependingon the values of its two parameters The variety of shapes ofthe hazard function of the TN(120572 120573) enables it to be a goodmodel to fit different data sets

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 532024 14 pageshttpdxdoiorg1011552014532024

2 Journal of Probability and Statistics

The rest of the paper is organized as follows Section 2introduces the new one-parameter distribution and someof its characteristics are discussed in Section 3 Section 4presents the transformation of the new distributionTN(120572 120573) Different characteristics of TN(120572 120573) such asthe hazard function quantiles random sample generationmoments and order statistics distributions are discussedin Section 5 Section 6 discusses the maximum likelihoodestimate of the two parameters of TN(120572 120573) Applicationsof the two models are presented in Section 6 Monte CarloSimulation study is carried out in Section 7 to examinethe accuracy of the maximum likelihood estimators of theTN(120572 120573) parameters as well as the coverage probability andaverage width of the confidence intervals for the parametersFinally Section 8 concludes this paper

2 The New Distribution

Consider the random variable 119879 whose pdf is given by

119891 (119905) =120573

1 + 120573[120573 + (1 + 2120573119905) 119890minus120573119905] 119890minus120573119905 119905 ge 0 120573 gt 0 (3)

The survival function (sf) of 119879 is given by

119878 (119905) =1

1 + 120573[120573 + (1 + 120573119905) 119890minus120573119905] 119890minus120573119905 119905 ge 0 120573 gt 0 (4)

while its hazard rate function is given by

ℎ (119905) =120573 [120573 + (1 + 2120573119909) 119890minus120573119905]

120573 + (1 + 120573119905) 119890minus120573119905 119905 ge 0 120573 gt 0 (5)

For simplicity from now on we refer to this distribution as119873(120573)

Interpretation There are two different interpretationsof 119873(120573) as follows

(1) The pdf 119891(119905) is a mixture density of two mixturecomponents One follows Exp(120573) and the other isthe lifetime of a two independent component seriessystem with Exp(120573) and 119866(2 120573) and mixture weights1198861= 120573(1 + 120573) and 119886

2= 1(1 + 120573) respec-

tively This means that 119873(120573) can be expressed interms of Exp(120573) and 119866(2 120573) as 119873(120573) = 119886

1Exp(120573) +

1198862minExp(120573) 119866(2 120573)

(2) The random variable 119879 can be described as a mix-ture of three components Exp(120573) Exp(2120573) anda 119866(2 2120573) with mixture weights 119887

1= 120573(1 +

120573) 1198872= 12(1 + 120573) and 119887

3= 12(1 + 120573) respec-

tively This means that 119873(120573) can be expressed interms of Exp(120573) Exp(2120573) and 119866(2 2120573) as 119873(120573) =1198871Exp(120573) + 119887

2Exp(2120573) + 119887

3119866(2 2120573)

Some characteristics of 119873(120573) are derived in the next section

3 Characteristics of119873(120573)

In this section algorithms are described to obtain quan-tiles of 119873(120573) and to generate samples from 119873(120573) Also

the moment generating function and the moments of thisdistribution are derived

31 Quantiles The 100119902th quantile 119902 isin (0 1) can be derivedas follows

(1) Let 119902 isin (0 1)

(2) Solve the following equation numerically in 119906 isin(0 1)

[120573 + (1 minus ln 119906) 119906] 119906 minus (1 + 120573) (1 minus 119902) = 0 (6)

(3) The 100119902th quantile is 119905119902= minus ln 119906120573

32 Random Sample Generation We provide belowthree equivalent algorithms to generate a random variatefrom 119873(120573)

Algorithm 1 (1) Generate 119906 sim 119880(0 1)(2) Solve numerically the following equation in V isin

(0 1) [120573 + (1 minus ln V)V]V minus (1 + 120573)119906 = 0(3) Set 119879 = minus ln V120573

Algorithm 2 (1) Generate I from the set 1 2 suchthat 119875(119868 = 119895) = 119886

119895 119895 = 1 2

(a) If 119868 = 1 set 1199101= minus ln 119906120573 where 119906 sim 119880(0 1)

(b) If 119868 = 2 set 1199102= minminus ln 119906

1120573 minus ln 119906

21199063120573

where 1199061 1199062 1199063sim 119880(0 1)

(2) Set 119879 = 119910119868

Algorithm 3 (1) Generate I from the set 1 2 3 suchthat 119875(119868 = 119895) = 119887

119895 119895 = 1 2 3

(a) If 119868 = 1 set 1199101= minus ln 119906120573 where 119906 sim 119880(0 1)

(b) If 119868 = 2 set 1199102= minus ln 1199062120573 where 119906 sim 119880(0 1)

(c) If 119868 = 3 set 1199103= minus ln 119906

111990622120573 where 119906

1 1199062sim

119880(0 1)

(2) Set 119879 = 119910119868

33 The Moments and the Moment Generating Function Themoment generating function (mgf) of 119873(120573) may be writtenas

119872(119905) =120573

1 + 120573[120573

120573 minus 119905+4120573 minus 119905

(2120573 minus 119905)2] 119905 lt 120573 (7)

Differentiating the above expression 119896 times with respect to 119905and setting 119905 to zero we get 119896th moments 120583

119896 as

120583119896= 119864 (119879119896) =

120573

1 + 120573[1 +

120573

2120573(1 +

119896 + 1

2120573)1

2119896]119896

120573119896 119896 ge 1

(8)

Journal of Probability and Statistics 3

0 1 2 3 4 5

120573

0 1 2 3 4 5

120573

185

19

195

2

205

21

215

22Sk

ewne

ss

88

9

92

94

96

98

10

102

104

106

108

Kurt

osis

Figure 1 Skewness and kurtosis of119873(120573) as functions 120573

Based on the first four ordinarymoments themeasures ofskewness (sk) and kurtosis (120581) of119873(120573) can be obtained using

sk =1205833minus 312058311205832+ 212058331

[1205832minus 12058321]32

(9)

120581 =1205834minus 412058311205833+ 6120583211205832minus 312058341

[1205832minus 12058321]2

(10)

Plots of the skewness and kurtosis of the distribution asa function of 120573 are plotted in Figure 1 From the plots skand 120581 are unimodal functions of 120573 The skewness is alwayspositive and the kurtosis is larger than 3 therefore 119873(120573) ispositively skewed and leptokurtic

4 Power Transformation ofthe New Distribution

To get a more flexible distribution we consider an extensionof the new distribution 119873(120573) with the pdf (3) by using thepower transformation119883 = 1198791120572 120572 gt 0 The pdf of119883 is givenby

119891 (119909) =120572120573119909120572minus1

1 + 120573[120573 + (1 + 2120573119909120572) 119890minus120573119909

120572

] 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(11)

The density of 119883 is plotted in Figure 2 for three choices of 120572when 120573 = 10 which shows that the density is symmetricwhen 120572 = 3535 left skewed when 120572 lt 3535 andright skewed when 120572 gt 3535 This implies that the power

parameter 120572 characterizes the shape of the density functionMore investigation of the density will be discussed in thenext section based on the skewness and kurtosis measuresFrom now on we will use TN(120572 120573) to refer to the powertransformation of the new distribution119873(120573)

Interpretation There are two different interpretations of 119884 asfollows

(1) The pdf 119891(119909) is a mixture density of two mixturecomponents One component follows 119882(120572 120573) andthe other is the lifetime of a two independent compo-nent series system with119882(120572 2120573) and PG(120572 2 120573) andmixture weights 119886

1= 120573(1 + 120573) and 119886

2= 1(1 + 120573)

respectively

(2) The random variable 119883 can be described as a mix-ture of three components 119882(120572 120573) 119882(120572 2120573) and aPG(120572 2 2120573)withmixture weights 119887

1= 120573(1+120573) 119887

2=

12(1 + 120573) and 1198873= 12(1 + 120573) respectively

Straightforward calculations yield the the survival function ofTN(120572 120573) as

119878 (119909) =1

1 + 120573[120573 + (1 + 120573119909120572) 119890minus120573119909

120572

] 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(12)

We derive some characteristics of TN(120572 120573) in the nextsection

4 Journal of Probability and Statistics

0

05

1

15

2

25

3

35

4

Den

sity

0 05 1 15 2 25 3

x

120572 = 11

120572 = 3535

120572 = 5

Figure 2 The TN(120572 120573) density function for some values of 120572 when120573 = 1

5 Characteristics of TN(120572 120573)51 The Hazard Function The hazard rate function ofTN(120572 120573) is

ℎ (119909) =120572120573119909120572minus1 [120573 + (1 + 2120573119909120572) 119890minus120573119909

120572

]

120573 + (1 + 120573119909120572) 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(13)

For120572 = 1 the hazard function is unimodal Its limiting valuesat zero and infinity are 120573 and it reaches a maximum value of

(1 + 2119882minus1(119890minus1120573)) 120573

1 +119882minus1(119890minus1120573)

(14)

at

1

120573(1 +119882

minus1(119890minus1

120573)) (15)

where119882minus1(sdot) denotes the Lambert119882 function which is the

inverse of the function 119909119890119909For 120572 = 1 the shape of the hazard function is difficult to

ascertain analyticallyThe shape was determined numericallyby examining the derivative of the hazard out to the 9999thpercentile of the distribution and the results are shown inFigure 3 For 120572 lt 1 the hazard is decreasing except fora small region with 120572 close to 1 and 120573 lt 05 where thehazard is initially decreasing then increasing and finallydecreasing (DID) For 120572 gt 1 the hazard is strictly increasingfor large 120573 (120573 gt 26 in the figure) For smaller 120573 with 120572close to 1 the hazard can be unimodal (for very small 120573) orinitially increasing then decreasing and finally increasing(IDI) (for slightly larger 120573) Figure 4 shows the hazard forfive choices of 120572 and 120573 which demonstrate the five possibleshapes

00 05 10 15 20 25 30 35

120572

n n n n00

05

10

15

20

25

30

35

120573

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minus

minusminusminusminusminusminusminusminusminus

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++

+

++++++

++++++++++++++++++++++++++

++++++++

++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

+++++++++++++++++

+++++++

++++++++

+++++++++++++++++++

+++++

+

++

+++++++++

++++++++++++++++++

++++++

+++++++++++++++++++++++++++

++++++

+++++++++++++++++++++++++++

+++++

++++++++++++++++++++++++++++

+++

+++++++++++++++

++

+

++

++++++

+

+++++++++

++

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

n

nn

iiiiiiiiiiiiiiiiii

iiiiiiiiiii

i

iiiiiiiii ii ii ii i i i i

iiiiiddd

d

d

DecreasingIncreasingUnimodal

IDIDID

Figure 3 Shapes of the hazard function of TN(120572 120573)

52 Quantiles and Random Sample Generations The 100119902thquantile of TN(120572 120573) 119909

119902 can be derived from that of119873(120573) 119905

119902

as follows

119909119902= 1199051120572119902 (16)

Figure 5 depicts the three quartiles11987611198762 and119876

3 which can

be obtained from the 119902th quantile by setting 119902 = 025 050and 075 respectively From Figure 5 the Interquartile range(IQR = 119876

3minus 1198761) decreases dramatically when 120572 increases

The following algorithm generates a random variate fromTN(120572 120573)

Algorithm 4 (1) Generate 119879 from 119873(120573) using one of theAlgorithms 1ndash3

(2) Set119883 = 1198791120572

53 The Moments and Shape Measures Let 119883 followTN(120572 120573) After some algebra the 119896th ordinary moment of119883is derived as

120583119896= 119864 (119883119896)

= 1198871

Γ (119896120572 + 1)

120573119896120572+ 1198871

Γ (119896120572 + 1)

(2120573)119896120572

+ 1198873

Γ (119896120572 + 2)

(2120573)119896120572

=119896Γ (119896120572) [2120572 (1 + 2119896120572120573) + 119896]

21205722 (1 + 120573) (2120573)119896120572

(17)

Journal of Probability and Statistics 5

Therefore the mean and variance of119883 are

120583 =Γ (1120572) [2120572 (1 + 21120572120573) + 1]

21205722 (1 + 120573) (2120573)1120572

1205902 = (81205722 (1 + 120573) Γ (2

120572) [120572 (1 + 22120572120573) + 1] minus Γ2 (

1

120572)

times[2120572 (1 + 21120572120573) + 1]2

)

times (41205724(1 + 120573)2

(2120573)2120572

)minus1

(18)

Figure 6 depicts the mean and variance of TN(120572 120573) asfunctions of 120572 when 120573 = 1 which shows that the meandecreases dramatically in 120572 and takes its minimum of 08298at 120572 = 161 then it increases steadily to take its maximum of09687 while the variance is decreasing

Based on the first four ordinary moments the measuresof skewness (sk) and kurtosis (120581) of TN(120572 120573) can be obtainedby substituting (17) into (9) and (10) respectively Plots of theskewness and kurtosis of TN(120572 120573) distribution as functionsof 120572 when 120573 = 10 are given in Figure 7 From these plots(1) the skewness is positive when 120572 lt 3535 and negativewhen 120572 gt 3535 and the kurtosis is (i) equal to 3 when either120572 = 24977 or 120572 = 49151 which means that the distributionis mesokurtic (ii) greater than 3 when either 120572 lt 24977 or120572 gt 49151 which means that the distribution is leptokurtic(iii) smaller than 3 when 24977 lt 120572 lt 49151 which meansthat the distribution is platykurtic This analysis shows howthe power parameter 120572 improves 119873(120573) because the powertransformationmodel can be used for datawith awide varietyof distributional shapes

54 Order Statistics Consider 119899 independent and identi-cal components whose lifetimes say 119883

1 1198832 119883

119899 fol-

low TN(120572 120573) The following theorem gives the limit-ing distributions of the lifetime of the series system1198831119899= min119883

1 119883

119899 and of the parallel system 119883

119899119899=

max1198831 119883

119899 consisting of these 119899 components

Theorem 5 The limiting distributions of1198831119899

and119883119899119899

are

lim119899rarrinfin

1198751198831119899minus 119886119899

119887119899

le 119909 = 1 minus 119890minus119909120572

119909 gt 0 (19)

lim119899rarrinfin

119875119883119899119899minus 119888119899

119889119899

le 119909 = exp minus exp (minus119909)

119909 isin (minusinfininfin)

(20)

where 119886119899= 0 119887119899= 119865minus1(1119899) 119888

119899= 119865minus1(1 minus 1119899) 119889

119899= 1119899119891(119888

119899)

and 119865(119909) = 1 minus 119878(119909) is the cdf of 119883

Proof Using LrsquoHospital rule

lim119905rarr0+

119865 (119905119909)

119865 (119905)= lim119905rarr0+

119909119891 (119905119909)

119891 (119905)

= lim119905rarr0+

119909120572 [120573 + (1 + 2120573(119905119909)120572) 119890minus120573(119905119909)120572

] 119890minus120573(119905119909)120572

[120573 + (1 + 2120573119905120572) 119890minus120573119905120572

] 119890minus120573119905120572

= 119909120572

(21)

Therefore (19) follows by Theorem 836(ii) of Arnold et al[4]

For the power transformation we have

lim119909rarrinfin

119889

119889119909

1

ℎ (119909)

= lim119909rarrinfin

minus(120572 minus 1) (120573 + (1 + 120573119909120572) 119890minus120573119909

120572

)

120572120573119909120572 (120573 + (1 + 2 120573119909120572) 119890minus120573119909120572

)

+ lim119909rarrinfin

120573119909120572

1 + 2120573119909120572 + 120573119890120573119909120572

+ lim119909rarrinfin

120573 (2120573119909120572 minus 1) + (120573119909120572 (1 + 2120573119909120572) minus 1) 119890minus120573119909120572

[120573 + (1 + 2120573119909120572) 119890minus120573119909120572

]2

119890120573119909120572

= 0

(22)

Therefore (20) follows by Theorem 833 of Arnold et al [4]

The following Theorem gives the limiting distribution ofthe 119894th order statistic of the 119899 lifetimes119883

1 1198832 119883

119899

Theorem 6 The limiting distributions of119883119894119899 1 le 119894 le 119899 are

lim119899rarrinfin

119875119883119894119899minus 119886119899

119887119899

le 119909 = 1 minus119894minus1

sum119895=0

119909119895120572

119895119890minus119909120572

119909 gt 0 (23)

where 119886119899= 0 119887119899= 119865minus1(1119899)

Proof It follows fromTheorem 5 and (842) of Arnold et al[4]

Theorem 5means that (1198831119899minus119886119899)119887119899and (119883

119899119899minus119888119899)119889119899fol-

low asymptotically119882(1 120572) and Ext(0 1) respectively whileTheorem 6 means that (119883

119894119899minus 119886119899)119887119899follows asymptotically

PG(120572 119894 1)

6 Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is one of the mostcommon methods for estimating the parameters of a sta-tistical model Assume that 119899 independent and identicalitems whose lifetimes follow TN(120572 120573) are put on a lifetest simultaneously Let 119909

1 1199092 119909

119899be the failure times of

6 Journal of Probability and Statistics

0 10 20 30 40

00

05

10

15

x

Haz

ard

(a) 120572 = 05 120573 = 01

x

Haz

ard

0 20 40 60 80 100

006

010

014

018

(b) 120572 = 09 120573 = 01

Haz

ard

x

0 20 40 60 80 100

010

012

014

(c) 120572 = 1 120573 = 01

Haz

ard

x

0 2 4 6 8 10

02

04

06

08

10

(d) 120572 = 12 120573 = 05

Haz

ard

x

0 1 2 3 4 5

00

10

20

30

(e) 120572 = 25 120573 = 01

Figure 4 The hazard function of TN(120572 120573) for some parameter values

Journal of Probability and Statistics 7

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

120572

Q1

Q2

Q3

Figure 5 The quartiles of TN(120572 120573) as functions of 120572 when 120573 = 1

the items and let x = (1199091 119909

119899)The likelihood function for

(120572 120573) is

119871 (120572 120573 x) = 120572119899120573119899

(1 + 120573)119899(119899

prod119894=1

119909119894)

120572minus1

119890minus120573sum119899

119894=1119909120572

119894

times119899

prod119894=1

[120573 + (1 + 2120573119909120572119894) 119890minus120573119909

120572

119894 ]

(24)

The log-likelihood function is

L (120572 120573 x) = 119899 ln120572 + 119899 ln120573 minus 119899 ln (1 + 120573)

+ (120572 minus 1)119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894+119899

sum119894=1

ln119860119894(120572 120573)

(25)

where

119860119894(120572 120573) = 120573 + (1 + 2120573119909120572

119894) 119890minus120573119909

120572

119894 119894 = 1 119899 (26)

Thefirst partial derivatives of L with respect to 120572 and 120573 are

L120572=119899

120572+119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894ln119909119894+119899

sum119894=1

119860119894120572(120572 120573)

119860119894(120572 120573)

L120573=

119899

120573 (1 + 120573)minus119899

sum119894=1

119909120572119894+119899

sum119894=1

119860119894120573(120572 120573)

119860119894(120572 120573)

(27)

where

119860119894120572(120572 120573) =

120597119860119894(120572 120573)

120597120572= 120573119909120572119894(1 minus 2120573119909120572

119894) ln119909119894119890minus120573119909

120572

119894

119860119894120573(120572 120573) =

120597119860119894(120572 120573)

120597120573= 1 + (1 minus 2120573119909120572

119894) 119909120572119894119890minus120573119909

120572

119894

(28)

The second partial derivatives ofL are

L120572120572= minus119899

1205722minus 120573119899

sum119894=1

119909120572119894(ln119909119894)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205722 (120572 120573) minus [119860

119894120572(120572 120573)]

2

[119860119894(120572 120573)]

2

L120572120573= minus119899

sum119894=1

119909120572119894ln119909119894

+119899

sum119894=1

119860119894(120572 120573)119860

119894120572120573(120572 120573) minus 119860

119894120572(120572 120573)119860

119894120573(120572 120573)

[119860119894(120572 120573)]

2

L120573120573= minus119899

1205732+

119899

(1 + 120573)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205732 (120572 120573) minus [119860

119894120573(120572 120573)]

2

[119860119894(120572 120573)]

2

(29)where1198601198941205722 (120572 120573) = 120573119909120572

119894(ln119909119894)2

119890minus120573119909120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

119860119894120572120573(120572 120573) = 119909120572

119894(ln119909119894) 119890minus120573119909

120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

1198601198941205732 (120572 120573) = minus1199092120572

119894119890minus120573119909

120572

119894 (3 minus 2120573119909120572119894)

(30)

The information matrix is

F (120572 120573) = minus [L120572120572

L120572120573

L120572120573

L120573120573

] (31)

The MLE of 120572 and 120573 say and 120573 are the solution of thesystem of nonlinear equations obtained by setting L

120572=

0 and L120573= 0 such that the F( 120573) is positive definite

This system has no analytic solution so numerical methodssuch as the Newton-Raphsonmethod Burden and Faires [5]should be used

Large-Sample Intervals The MLE of the parameters 120572and 120573 are asymptotically normally distributed with meansequal to the true values of 120572 and 120573 and variances given bythe inverse of the information matrix In particular

(

120573) sim 119873

2((120572120573) Fminus1) (32)

where Fminus1 is the inverse of F( 120573) with main diagonalelements F11 and F22 given by

F11 =

F22

F11F22minus F212

F22 =

F11

F11F22minus F212

(33)

Using (32) large-sample (1 minus 120599)100 confidence intervalsfor 120572 and 120573 are

plusmn 1198851205992radicF11 120573 plusmn 119885

1205992radicF22 (34)

where 1198851205992

is the upper 1001205992 quantile of the standardnormal distribution

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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Stochastic AnalysisInternational Journal of

2 Journal of Probability and Statistics

The rest of the paper is organized as follows Section 2introduces the new one-parameter distribution and someof its characteristics are discussed in Section 3 Section 4presents the transformation of the new distributionTN(120572 120573) Different characteristics of TN(120572 120573) such asthe hazard function quantiles random sample generationmoments and order statistics distributions are discussedin Section 5 Section 6 discusses the maximum likelihoodestimate of the two parameters of TN(120572 120573) Applicationsof the two models are presented in Section 6 Monte CarloSimulation study is carried out in Section 7 to examinethe accuracy of the maximum likelihood estimators of theTN(120572 120573) parameters as well as the coverage probability andaverage width of the confidence intervals for the parametersFinally Section 8 concludes this paper

2 The New Distribution

Consider the random variable 119879 whose pdf is given by

119891 (119905) =120573

1 + 120573[120573 + (1 + 2120573119905) 119890minus120573119905] 119890minus120573119905 119905 ge 0 120573 gt 0 (3)

The survival function (sf) of 119879 is given by

119878 (119905) =1

1 + 120573[120573 + (1 + 120573119905) 119890minus120573119905] 119890minus120573119905 119905 ge 0 120573 gt 0 (4)

while its hazard rate function is given by

ℎ (119905) =120573 [120573 + (1 + 2120573119909) 119890minus120573119905]

120573 + (1 + 120573119905) 119890minus120573119905 119905 ge 0 120573 gt 0 (5)

For simplicity from now on we refer to this distribution as119873(120573)

Interpretation There are two different interpretationsof 119873(120573) as follows

(1) The pdf 119891(119905) is a mixture density of two mixturecomponents One follows Exp(120573) and the other isthe lifetime of a two independent component seriessystem with Exp(120573) and 119866(2 120573) and mixture weights1198861= 120573(1 + 120573) and 119886

2= 1(1 + 120573) respec-

tively This means that 119873(120573) can be expressed interms of Exp(120573) and 119866(2 120573) as 119873(120573) = 119886

1Exp(120573) +

1198862minExp(120573) 119866(2 120573)

(2) The random variable 119879 can be described as a mix-ture of three components Exp(120573) Exp(2120573) anda 119866(2 2120573) with mixture weights 119887

1= 120573(1 +

120573) 1198872= 12(1 + 120573) and 119887

3= 12(1 + 120573) respec-

tively This means that 119873(120573) can be expressed interms of Exp(120573) Exp(2120573) and 119866(2 2120573) as 119873(120573) =1198871Exp(120573) + 119887

2Exp(2120573) + 119887

3119866(2 2120573)

Some characteristics of 119873(120573) are derived in the next section

3 Characteristics of119873(120573)

In this section algorithms are described to obtain quan-tiles of 119873(120573) and to generate samples from 119873(120573) Also

the moment generating function and the moments of thisdistribution are derived

31 Quantiles The 100119902th quantile 119902 isin (0 1) can be derivedas follows

(1) Let 119902 isin (0 1)

(2) Solve the following equation numerically in 119906 isin(0 1)

[120573 + (1 minus ln 119906) 119906] 119906 minus (1 + 120573) (1 minus 119902) = 0 (6)

(3) The 100119902th quantile is 119905119902= minus ln 119906120573

32 Random Sample Generation We provide belowthree equivalent algorithms to generate a random variatefrom 119873(120573)

Algorithm 1 (1) Generate 119906 sim 119880(0 1)(2) Solve numerically the following equation in V isin

(0 1) [120573 + (1 minus ln V)V]V minus (1 + 120573)119906 = 0(3) Set 119879 = minus ln V120573

Algorithm 2 (1) Generate I from the set 1 2 suchthat 119875(119868 = 119895) = 119886

119895 119895 = 1 2

(a) If 119868 = 1 set 1199101= minus ln 119906120573 where 119906 sim 119880(0 1)

(b) If 119868 = 2 set 1199102= minminus ln 119906

1120573 minus ln 119906

21199063120573

where 1199061 1199062 1199063sim 119880(0 1)

(2) Set 119879 = 119910119868

Algorithm 3 (1) Generate I from the set 1 2 3 suchthat 119875(119868 = 119895) = 119887

119895 119895 = 1 2 3

(a) If 119868 = 1 set 1199101= minus ln 119906120573 where 119906 sim 119880(0 1)

(b) If 119868 = 2 set 1199102= minus ln 1199062120573 where 119906 sim 119880(0 1)

(c) If 119868 = 3 set 1199103= minus ln 119906

111990622120573 where 119906

1 1199062sim

119880(0 1)

(2) Set 119879 = 119910119868

33 The Moments and the Moment Generating Function Themoment generating function (mgf) of 119873(120573) may be writtenas

119872(119905) =120573

1 + 120573[120573

120573 minus 119905+4120573 minus 119905

(2120573 minus 119905)2] 119905 lt 120573 (7)

Differentiating the above expression 119896 times with respect to 119905and setting 119905 to zero we get 119896th moments 120583

119896 as

120583119896= 119864 (119879119896) =

120573

1 + 120573[1 +

120573

2120573(1 +

119896 + 1

2120573)1

2119896]119896

120573119896 119896 ge 1

(8)

Journal of Probability and Statistics 3

0 1 2 3 4 5

120573

0 1 2 3 4 5

120573

185

19

195

2

205

21

215

22Sk

ewne

ss

88

9

92

94

96

98

10

102

104

106

108

Kurt

osis

Figure 1 Skewness and kurtosis of119873(120573) as functions 120573

Based on the first four ordinarymoments themeasures ofskewness (sk) and kurtosis (120581) of119873(120573) can be obtained using

sk =1205833minus 312058311205832+ 212058331

[1205832minus 12058321]32

(9)

120581 =1205834minus 412058311205833+ 6120583211205832minus 312058341

[1205832minus 12058321]2

(10)

Plots of the skewness and kurtosis of the distribution asa function of 120573 are plotted in Figure 1 From the plots skand 120581 are unimodal functions of 120573 The skewness is alwayspositive and the kurtosis is larger than 3 therefore 119873(120573) ispositively skewed and leptokurtic

4 Power Transformation ofthe New Distribution

To get a more flexible distribution we consider an extensionof the new distribution 119873(120573) with the pdf (3) by using thepower transformation119883 = 1198791120572 120572 gt 0 The pdf of119883 is givenby

119891 (119909) =120572120573119909120572minus1

1 + 120573[120573 + (1 + 2120573119909120572) 119890minus120573119909

120572

] 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(11)

The density of 119883 is plotted in Figure 2 for three choices of 120572when 120573 = 10 which shows that the density is symmetricwhen 120572 = 3535 left skewed when 120572 lt 3535 andright skewed when 120572 gt 3535 This implies that the power

parameter 120572 characterizes the shape of the density functionMore investigation of the density will be discussed in thenext section based on the skewness and kurtosis measuresFrom now on we will use TN(120572 120573) to refer to the powertransformation of the new distribution119873(120573)

Interpretation There are two different interpretations of 119884 asfollows

(1) The pdf 119891(119909) is a mixture density of two mixturecomponents One component follows 119882(120572 120573) andthe other is the lifetime of a two independent compo-nent series system with119882(120572 2120573) and PG(120572 2 120573) andmixture weights 119886

1= 120573(1 + 120573) and 119886

2= 1(1 + 120573)

respectively

(2) The random variable 119883 can be described as a mix-ture of three components 119882(120572 120573) 119882(120572 2120573) and aPG(120572 2 2120573)withmixture weights 119887

1= 120573(1+120573) 119887

2=

12(1 + 120573) and 1198873= 12(1 + 120573) respectively

Straightforward calculations yield the the survival function ofTN(120572 120573) as

119878 (119909) =1

1 + 120573[120573 + (1 + 120573119909120572) 119890minus120573119909

120572

] 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(12)

We derive some characteristics of TN(120572 120573) in the nextsection

4 Journal of Probability and Statistics

0

05

1

15

2

25

3

35

4

Den

sity

0 05 1 15 2 25 3

x

120572 = 11

120572 = 3535

120572 = 5

Figure 2 The TN(120572 120573) density function for some values of 120572 when120573 = 1

5 Characteristics of TN(120572 120573)51 The Hazard Function The hazard rate function ofTN(120572 120573) is

ℎ (119909) =120572120573119909120572minus1 [120573 + (1 + 2120573119909120572) 119890minus120573119909

120572

]

120573 + (1 + 120573119909120572) 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(13)

For120572 = 1 the hazard function is unimodal Its limiting valuesat zero and infinity are 120573 and it reaches a maximum value of

(1 + 2119882minus1(119890minus1120573)) 120573

1 +119882minus1(119890minus1120573)

(14)

at

1

120573(1 +119882

minus1(119890minus1

120573)) (15)

where119882minus1(sdot) denotes the Lambert119882 function which is the

inverse of the function 119909119890119909For 120572 = 1 the shape of the hazard function is difficult to

ascertain analyticallyThe shape was determined numericallyby examining the derivative of the hazard out to the 9999thpercentile of the distribution and the results are shown inFigure 3 For 120572 lt 1 the hazard is decreasing except fora small region with 120572 close to 1 and 120573 lt 05 where thehazard is initially decreasing then increasing and finallydecreasing (DID) For 120572 gt 1 the hazard is strictly increasingfor large 120573 (120573 gt 26 in the figure) For smaller 120573 with 120572close to 1 the hazard can be unimodal (for very small 120573) orinitially increasing then decreasing and finally increasing(IDI) (for slightly larger 120573) Figure 4 shows the hazard forfive choices of 120572 and 120573 which demonstrate the five possibleshapes

00 05 10 15 20 25 30 35

120572

n n n n00

05

10

15

20

25

30

35

120573

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minus

minusminusminusminusminusminusminusminusminus

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++

+

++++++

++++++++++++++++++++++++++

++++++++

++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

+++++++++++++++++

+++++++

++++++++

+++++++++++++++++++

+++++

+

++

+++++++++

++++++++++++++++++

++++++

+++++++++++++++++++++++++++

++++++

+++++++++++++++++++++++++++

+++++

++++++++++++++++++++++++++++

+++

+++++++++++++++

++

+

++

++++++

+

+++++++++

++

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

n

nn

iiiiiiiiiiiiiiiiii

iiiiiiiiiii

i

iiiiiiiii ii ii ii i i i i

iiiiiddd

d

d

DecreasingIncreasingUnimodal

IDIDID

Figure 3 Shapes of the hazard function of TN(120572 120573)

52 Quantiles and Random Sample Generations The 100119902thquantile of TN(120572 120573) 119909

119902 can be derived from that of119873(120573) 119905

119902

as follows

119909119902= 1199051120572119902 (16)

Figure 5 depicts the three quartiles11987611198762 and119876

3 which can

be obtained from the 119902th quantile by setting 119902 = 025 050and 075 respectively From Figure 5 the Interquartile range(IQR = 119876

3minus 1198761) decreases dramatically when 120572 increases

The following algorithm generates a random variate fromTN(120572 120573)

Algorithm 4 (1) Generate 119879 from 119873(120573) using one of theAlgorithms 1ndash3

(2) Set119883 = 1198791120572

53 The Moments and Shape Measures Let 119883 followTN(120572 120573) After some algebra the 119896th ordinary moment of119883is derived as

120583119896= 119864 (119883119896)

= 1198871

Γ (119896120572 + 1)

120573119896120572+ 1198871

Γ (119896120572 + 1)

(2120573)119896120572

+ 1198873

Γ (119896120572 + 2)

(2120573)119896120572

=119896Γ (119896120572) [2120572 (1 + 2119896120572120573) + 119896]

21205722 (1 + 120573) (2120573)119896120572

(17)

Journal of Probability and Statistics 5

Therefore the mean and variance of119883 are

120583 =Γ (1120572) [2120572 (1 + 21120572120573) + 1]

21205722 (1 + 120573) (2120573)1120572

1205902 = (81205722 (1 + 120573) Γ (2

120572) [120572 (1 + 22120572120573) + 1] minus Γ2 (

1

120572)

times[2120572 (1 + 21120572120573) + 1]2

)

times (41205724(1 + 120573)2

(2120573)2120572

)minus1

(18)

Figure 6 depicts the mean and variance of TN(120572 120573) asfunctions of 120572 when 120573 = 1 which shows that the meandecreases dramatically in 120572 and takes its minimum of 08298at 120572 = 161 then it increases steadily to take its maximum of09687 while the variance is decreasing

Based on the first four ordinary moments the measuresof skewness (sk) and kurtosis (120581) of TN(120572 120573) can be obtainedby substituting (17) into (9) and (10) respectively Plots of theskewness and kurtosis of TN(120572 120573) distribution as functionsof 120572 when 120573 = 10 are given in Figure 7 From these plots(1) the skewness is positive when 120572 lt 3535 and negativewhen 120572 gt 3535 and the kurtosis is (i) equal to 3 when either120572 = 24977 or 120572 = 49151 which means that the distributionis mesokurtic (ii) greater than 3 when either 120572 lt 24977 or120572 gt 49151 which means that the distribution is leptokurtic(iii) smaller than 3 when 24977 lt 120572 lt 49151 which meansthat the distribution is platykurtic This analysis shows howthe power parameter 120572 improves 119873(120573) because the powertransformationmodel can be used for datawith awide varietyof distributional shapes

54 Order Statistics Consider 119899 independent and identi-cal components whose lifetimes say 119883

1 1198832 119883

119899 fol-

low TN(120572 120573) The following theorem gives the limit-ing distributions of the lifetime of the series system1198831119899= min119883

1 119883

119899 and of the parallel system 119883

119899119899=

max1198831 119883

119899 consisting of these 119899 components

Theorem 5 The limiting distributions of1198831119899

and119883119899119899

are

lim119899rarrinfin

1198751198831119899minus 119886119899

119887119899

le 119909 = 1 minus 119890minus119909120572

119909 gt 0 (19)

lim119899rarrinfin

119875119883119899119899minus 119888119899

119889119899

le 119909 = exp minus exp (minus119909)

119909 isin (minusinfininfin)

(20)

where 119886119899= 0 119887119899= 119865minus1(1119899) 119888

119899= 119865minus1(1 minus 1119899) 119889

119899= 1119899119891(119888

119899)

and 119865(119909) = 1 minus 119878(119909) is the cdf of 119883

Proof Using LrsquoHospital rule

lim119905rarr0+

119865 (119905119909)

119865 (119905)= lim119905rarr0+

119909119891 (119905119909)

119891 (119905)

= lim119905rarr0+

119909120572 [120573 + (1 + 2120573(119905119909)120572) 119890minus120573(119905119909)120572

] 119890minus120573(119905119909)120572

[120573 + (1 + 2120573119905120572) 119890minus120573119905120572

] 119890minus120573119905120572

= 119909120572

(21)

Therefore (19) follows by Theorem 836(ii) of Arnold et al[4]

For the power transformation we have

lim119909rarrinfin

119889

119889119909

1

ℎ (119909)

= lim119909rarrinfin

minus(120572 minus 1) (120573 + (1 + 120573119909120572) 119890minus120573119909

120572

)

120572120573119909120572 (120573 + (1 + 2 120573119909120572) 119890minus120573119909120572

)

+ lim119909rarrinfin

120573119909120572

1 + 2120573119909120572 + 120573119890120573119909120572

+ lim119909rarrinfin

120573 (2120573119909120572 minus 1) + (120573119909120572 (1 + 2120573119909120572) minus 1) 119890minus120573119909120572

[120573 + (1 + 2120573119909120572) 119890minus120573119909120572

]2

119890120573119909120572

= 0

(22)

Therefore (20) follows by Theorem 833 of Arnold et al [4]

The following Theorem gives the limiting distribution ofthe 119894th order statistic of the 119899 lifetimes119883

1 1198832 119883

119899

Theorem 6 The limiting distributions of119883119894119899 1 le 119894 le 119899 are

lim119899rarrinfin

119875119883119894119899minus 119886119899

119887119899

le 119909 = 1 minus119894minus1

sum119895=0

119909119895120572

119895119890minus119909120572

119909 gt 0 (23)

where 119886119899= 0 119887119899= 119865minus1(1119899)

Proof It follows fromTheorem 5 and (842) of Arnold et al[4]

Theorem 5means that (1198831119899minus119886119899)119887119899and (119883

119899119899minus119888119899)119889119899fol-

low asymptotically119882(1 120572) and Ext(0 1) respectively whileTheorem 6 means that (119883

119894119899minus 119886119899)119887119899follows asymptotically

PG(120572 119894 1)

6 Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is one of the mostcommon methods for estimating the parameters of a sta-tistical model Assume that 119899 independent and identicalitems whose lifetimes follow TN(120572 120573) are put on a lifetest simultaneously Let 119909

1 1199092 119909

119899be the failure times of

6 Journal of Probability and Statistics

0 10 20 30 40

00

05

10

15

x

Haz

ard

(a) 120572 = 05 120573 = 01

x

Haz

ard

0 20 40 60 80 100

006

010

014

018

(b) 120572 = 09 120573 = 01

Haz

ard

x

0 20 40 60 80 100

010

012

014

(c) 120572 = 1 120573 = 01

Haz

ard

x

0 2 4 6 8 10

02

04

06

08

10

(d) 120572 = 12 120573 = 05

Haz

ard

x

0 1 2 3 4 5

00

10

20

30

(e) 120572 = 25 120573 = 01

Figure 4 The hazard function of TN(120572 120573) for some parameter values

Journal of Probability and Statistics 7

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

120572

Q1

Q2

Q3

Figure 5 The quartiles of TN(120572 120573) as functions of 120572 when 120573 = 1

the items and let x = (1199091 119909

119899)The likelihood function for

(120572 120573) is

119871 (120572 120573 x) = 120572119899120573119899

(1 + 120573)119899(119899

prod119894=1

119909119894)

120572minus1

119890minus120573sum119899

119894=1119909120572

119894

times119899

prod119894=1

[120573 + (1 + 2120573119909120572119894) 119890minus120573119909

120572

119894 ]

(24)

The log-likelihood function is

L (120572 120573 x) = 119899 ln120572 + 119899 ln120573 minus 119899 ln (1 + 120573)

+ (120572 minus 1)119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894+119899

sum119894=1

ln119860119894(120572 120573)

(25)

where

119860119894(120572 120573) = 120573 + (1 + 2120573119909120572

119894) 119890minus120573119909

120572

119894 119894 = 1 119899 (26)

Thefirst partial derivatives of L with respect to 120572 and 120573 are

L120572=119899

120572+119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894ln119909119894+119899

sum119894=1

119860119894120572(120572 120573)

119860119894(120572 120573)

L120573=

119899

120573 (1 + 120573)minus119899

sum119894=1

119909120572119894+119899

sum119894=1

119860119894120573(120572 120573)

119860119894(120572 120573)

(27)

where

119860119894120572(120572 120573) =

120597119860119894(120572 120573)

120597120572= 120573119909120572119894(1 minus 2120573119909120572

119894) ln119909119894119890minus120573119909

120572

119894

119860119894120573(120572 120573) =

120597119860119894(120572 120573)

120597120573= 1 + (1 minus 2120573119909120572

119894) 119909120572119894119890minus120573119909

120572

119894

(28)

The second partial derivatives ofL are

L120572120572= minus119899

1205722minus 120573119899

sum119894=1

119909120572119894(ln119909119894)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205722 (120572 120573) minus [119860

119894120572(120572 120573)]

2

[119860119894(120572 120573)]

2

L120572120573= minus119899

sum119894=1

119909120572119894ln119909119894

+119899

sum119894=1

119860119894(120572 120573)119860

119894120572120573(120572 120573) minus 119860

119894120572(120572 120573)119860

119894120573(120572 120573)

[119860119894(120572 120573)]

2

L120573120573= minus119899

1205732+

119899

(1 + 120573)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205732 (120572 120573) minus [119860

119894120573(120572 120573)]

2

[119860119894(120572 120573)]

2

(29)where1198601198941205722 (120572 120573) = 120573119909120572

119894(ln119909119894)2

119890minus120573119909120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

119860119894120572120573(120572 120573) = 119909120572

119894(ln119909119894) 119890minus120573119909

120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

1198601198941205732 (120572 120573) = minus1199092120572

119894119890minus120573119909

120572

119894 (3 minus 2120573119909120572119894)

(30)

The information matrix is

F (120572 120573) = minus [L120572120572

L120572120573

L120572120573

L120573120573

] (31)

The MLE of 120572 and 120573 say and 120573 are the solution of thesystem of nonlinear equations obtained by setting L

120572=

0 and L120573= 0 such that the F( 120573) is positive definite

This system has no analytic solution so numerical methodssuch as the Newton-Raphsonmethod Burden and Faires [5]should be used

Large-Sample Intervals The MLE of the parameters 120572and 120573 are asymptotically normally distributed with meansequal to the true values of 120572 and 120573 and variances given bythe inverse of the information matrix In particular

(

120573) sim 119873

2((120572120573) Fminus1) (32)

where Fminus1 is the inverse of F( 120573) with main diagonalelements F11 and F22 given by

F11 =

F22

F11F22minus F212

F22 =

F11

F11F22minus F212

(33)

Using (32) large-sample (1 minus 120599)100 confidence intervalsfor 120572 and 120573 are

plusmn 1198851205992radicF11 120573 plusmn 119885

1205992radicF22 (34)

where 1198851205992

is the upper 1001205992 quantile of the standardnormal distribution

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 3

0 1 2 3 4 5

120573

0 1 2 3 4 5

120573

185

19

195

2

205

21

215

22Sk

ewne

ss

88

9

92

94

96

98

10

102

104

106

108

Kurt

osis

Figure 1 Skewness and kurtosis of119873(120573) as functions 120573

Based on the first four ordinarymoments themeasures ofskewness (sk) and kurtosis (120581) of119873(120573) can be obtained using

sk =1205833minus 312058311205832+ 212058331

[1205832minus 12058321]32

(9)

120581 =1205834minus 412058311205833+ 6120583211205832minus 312058341

[1205832minus 12058321]2

(10)

Plots of the skewness and kurtosis of the distribution asa function of 120573 are plotted in Figure 1 From the plots skand 120581 are unimodal functions of 120573 The skewness is alwayspositive and the kurtosis is larger than 3 therefore 119873(120573) ispositively skewed and leptokurtic

4 Power Transformation ofthe New Distribution

To get a more flexible distribution we consider an extensionof the new distribution 119873(120573) with the pdf (3) by using thepower transformation119883 = 1198791120572 120572 gt 0 The pdf of119883 is givenby

119891 (119909) =120572120573119909120572minus1

1 + 120573[120573 + (1 + 2120573119909120572) 119890minus120573119909

120572

] 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(11)

The density of 119883 is plotted in Figure 2 for three choices of 120572when 120573 = 10 which shows that the density is symmetricwhen 120572 = 3535 left skewed when 120572 lt 3535 andright skewed when 120572 gt 3535 This implies that the power

parameter 120572 characterizes the shape of the density functionMore investigation of the density will be discussed in thenext section based on the skewness and kurtosis measuresFrom now on we will use TN(120572 120573) to refer to the powertransformation of the new distribution119873(120573)

Interpretation There are two different interpretations of 119884 asfollows

(1) The pdf 119891(119909) is a mixture density of two mixturecomponents One component follows 119882(120572 120573) andthe other is the lifetime of a two independent compo-nent series system with119882(120572 2120573) and PG(120572 2 120573) andmixture weights 119886

1= 120573(1 + 120573) and 119886

2= 1(1 + 120573)

respectively

(2) The random variable 119883 can be described as a mix-ture of three components 119882(120572 120573) 119882(120572 2120573) and aPG(120572 2 2120573)withmixture weights 119887

1= 120573(1+120573) 119887

2=

12(1 + 120573) and 1198873= 12(1 + 120573) respectively

Straightforward calculations yield the the survival function ofTN(120572 120573) as

119878 (119909) =1

1 + 120573[120573 + (1 + 120573119909120572) 119890minus120573119909

120572

] 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(12)

We derive some characteristics of TN(120572 120573) in the nextsection

4 Journal of Probability and Statistics

0

05

1

15

2

25

3

35

4

Den

sity

0 05 1 15 2 25 3

x

120572 = 11

120572 = 3535

120572 = 5

Figure 2 The TN(120572 120573) density function for some values of 120572 when120573 = 1

5 Characteristics of TN(120572 120573)51 The Hazard Function The hazard rate function ofTN(120572 120573) is

ℎ (119909) =120572120573119909120572minus1 [120573 + (1 + 2120573119909120572) 119890minus120573119909

120572

]

120573 + (1 + 120573119909120572) 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(13)

For120572 = 1 the hazard function is unimodal Its limiting valuesat zero and infinity are 120573 and it reaches a maximum value of

(1 + 2119882minus1(119890minus1120573)) 120573

1 +119882minus1(119890minus1120573)

(14)

at

1

120573(1 +119882

minus1(119890minus1

120573)) (15)

where119882minus1(sdot) denotes the Lambert119882 function which is the

inverse of the function 119909119890119909For 120572 = 1 the shape of the hazard function is difficult to

ascertain analyticallyThe shape was determined numericallyby examining the derivative of the hazard out to the 9999thpercentile of the distribution and the results are shown inFigure 3 For 120572 lt 1 the hazard is decreasing except fora small region with 120572 close to 1 and 120573 lt 05 where thehazard is initially decreasing then increasing and finallydecreasing (DID) For 120572 gt 1 the hazard is strictly increasingfor large 120573 (120573 gt 26 in the figure) For smaller 120573 with 120572close to 1 the hazard can be unimodal (for very small 120573) orinitially increasing then decreasing and finally increasing(IDI) (for slightly larger 120573) Figure 4 shows the hazard forfive choices of 120572 and 120573 which demonstrate the five possibleshapes

00 05 10 15 20 25 30 35

120572

n n n n00

05

10

15

20

25

30

35

120573

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minus

minusminusminusminusminusminusminusminusminus

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++

+

++++++

++++++++++++++++++++++++++

++++++++

++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

+++++++++++++++++

+++++++

++++++++

+++++++++++++++++++

+++++

+

++

+++++++++

++++++++++++++++++

++++++

+++++++++++++++++++++++++++

++++++

+++++++++++++++++++++++++++

+++++

++++++++++++++++++++++++++++

+++

+++++++++++++++

++

+

++

++++++

+

+++++++++

++

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

n

nn

iiiiiiiiiiiiiiiiii

iiiiiiiiiii

i

iiiiiiiii ii ii ii i i i i

iiiiiddd

d

d

DecreasingIncreasingUnimodal

IDIDID

Figure 3 Shapes of the hazard function of TN(120572 120573)

52 Quantiles and Random Sample Generations The 100119902thquantile of TN(120572 120573) 119909

119902 can be derived from that of119873(120573) 119905

119902

as follows

119909119902= 1199051120572119902 (16)

Figure 5 depicts the three quartiles11987611198762 and119876

3 which can

be obtained from the 119902th quantile by setting 119902 = 025 050and 075 respectively From Figure 5 the Interquartile range(IQR = 119876

3minus 1198761) decreases dramatically when 120572 increases

The following algorithm generates a random variate fromTN(120572 120573)

Algorithm 4 (1) Generate 119879 from 119873(120573) using one of theAlgorithms 1ndash3

(2) Set119883 = 1198791120572

53 The Moments and Shape Measures Let 119883 followTN(120572 120573) After some algebra the 119896th ordinary moment of119883is derived as

120583119896= 119864 (119883119896)

= 1198871

Γ (119896120572 + 1)

120573119896120572+ 1198871

Γ (119896120572 + 1)

(2120573)119896120572

+ 1198873

Γ (119896120572 + 2)

(2120573)119896120572

=119896Γ (119896120572) [2120572 (1 + 2119896120572120573) + 119896]

21205722 (1 + 120573) (2120573)119896120572

(17)

Journal of Probability and Statistics 5

Therefore the mean and variance of119883 are

120583 =Γ (1120572) [2120572 (1 + 21120572120573) + 1]

21205722 (1 + 120573) (2120573)1120572

1205902 = (81205722 (1 + 120573) Γ (2

120572) [120572 (1 + 22120572120573) + 1] minus Γ2 (

1

120572)

times[2120572 (1 + 21120572120573) + 1]2

)

times (41205724(1 + 120573)2

(2120573)2120572

)minus1

(18)

Figure 6 depicts the mean and variance of TN(120572 120573) asfunctions of 120572 when 120573 = 1 which shows that the meandecreases dramatically in 120572 and takes its minimum of 08298at 120572 = 161 then it increases steadily to take its maximum of09687 while the variance is decreasing

Based on the first four ordinary moments the measuresof skewness (sk) and kurtosis (120581) of TN(120572 120573) can be obtainedby substituting (17) into (9) and (10) respectively Plots of theskewness and kurtosis of TN(120572 120573) distribution as functionsof 120572 when 120573 = 10 are given in Figure 7 From these plots(1) the skewness is positive when 120572 lt 3535 and negativewhen 120572 gt 3535 and the kurtosis is (i) equal to 3 when either120572 = 24977 or 120572 = 49151 which means that the distributionis mesokurtic (ii) greater than 3 when either 120572 lt 24977 or120572 gt 49151 which means that the distribution is leptokurtic(iii) smaller than 3 when 24977 lt 120572 lt 49151 which meansthat the distribution is platykurtic This analysis shows howthe power parameter 120572 improves 119873(120573) because the powertransformationmodel can be used for datawith awide varietyof distributional shapes

54 Order Statistics Consider 119899 independent and identi-cal components whose lifetimes say 119883

1 1198832 119883

119899 fol-

low TN(120572 120573) The following theorem gives the limit-ing distributions of the lifetime of the series system1198831119899= min119883

1 119883

119899 and of the parallel system 119883

119899119899=

max1198831 119883

119899 consisting of these 119899 components

Theorem 5 The limiting distributions of1198831119899

and119883119899119899

are

lim119899rarrinfin

1198751198831119899minus 119886119899

119887119899

le 119909 = 1 minus 119890minus119909120572

119909 gt 0 (19)

lim119899rarrinfin

119875119883119899119899minus 119888119899

119889119899

le 119909 = exp minus exp (minus119909)

119909 isin (minusinfininfin)

(20)

where 119886119899= 0 119887119899= 119865minus1(1119899) 119888

119899= 119865minus1(1 minus 1119899) 119889

119899= 1119899119891(119888

119899)

and 119865(119909) = 1 minus 119878(119909) is the cdf of 119883

Proof Using LrsquoHospital rule

lim119905rarr0+

119865 (119905119909)

119865 (119905)= lim119905rarr0+

119909119891 (119905119909)

119891 (119905)

= lim119905rarr0+

119909120572 [120573 + (1 + 2120573(119905119909)120572) 119890minus120573(119905119909)120572

] 119890minus120573(119905119909)120572

[120573 + (1 + 2120573119905120572) 119890minus120573119905120572

] 119890minus120573119905120572

= 119909120572

(21)

Therefore (19) follows by Theorem 836(ii) of Arnold et al[4]

For the power transformation we have

lim119909rarrinfin

119889

119889119909

1

ℎ (119909)

= lim119909rarrinfin

minus(120572 minus 1) (120573 + (1 + 120573119909120572) 119890minus120573119909

120572

)

120572120573119909120572 (120573 + (1 + 2 120573119909120572) 119890minus120573119909120572

)

+ lim119909rarrinfin

120573119909120572

1 + 2120573119909120572 + 120573119890120573119909120572

+ lim119909rarrinfin

120573 (2120573119909120572 minus 1) + (120573119909120572 (1 + 2120573119909120572) minus 1) 119890minus120573119909120572

[120573 + (1 + 2120573119909120572) 119890minus120573119909120572

]2

119890120573119909120572

= 0

(22)

Therefore (20) follows by Theorem 833 of Arnold et al [4]

The following Theorem gives the limiting distribution ofthe 119894th order statistic of the 119899 lifetimes119883

1 1198832 119883

119899

Theorem 6 The limiting distributions of119883119894119899 1 le 119894 le 119899 are

lim119899rarrinfin

119875119883119894119899minus 119886119899

119887119899

le 119909 = 1 minus119894minus1

sum119895=0

119909119895120572

119895119890minus119909120572

119909 gt 0 (23)

where 119886119899= 0 119887119899= 119865minus1(1119899)

Proof It follows fromTheorem 5 and (842) of Arnold et al[4]

Theorem 5means that (1198831119899minus119886119899)119887119899and (119883

119899119899minus119888119899)119889119899fol-

low asymptotically119882(1 120572) and Ext(0 1) respectively whileTheorem 6 means that (119883

119894119899minus 119886119899)119887119899follows asymptotically

PG(120572 119894 1)

6 Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is one of the mostcommon methods for estimating the parameters of a sta-tistical model Assume that 119899 independent and identicalitems whose lifetimes follow TN(120572 120573) are put on a lifetest simultaneously Let 119909

1 1199092 119909

119899be the failure times of

6 Journal of Probability and Statistics

0 10 20 30 40

00

05

10

15

x

Haz

ard

(a) 120572 = 05 120573 = 01

x

Haz

ard

0 20 40 60 80 100

006

010

014

018

(b) 120572 = 09 120573 = 01

Haz

ard

x

0 20 40 60 80 100

010

012

014

(c) 120572 = 1 120573 = 01

Haz

ard

x

0 2 4 6 8 10

02

04

06

08

10

(d) 120572 = 12 120573 = 05

Haz

ard

x

0 1 2 3 4 5

00

10

20

30

(e) 120572 = 25 120573 = 01

Figure 4 The hazard function of TN(120572 120573) for some parameter values

Journal of Probability and Statistics 7

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

120572

Q1

Q2

Q3

Figure 5 The quartiles of TN(120572 120573) as functions of 120572 when 120573 = 1

the items and let x = (1199091 119909

119899)The likelihood function for

(120572 120573) is

119871 (120572 120573 x) = 120572119899120573119899

(1 + 120573)119899(119899

prod119894=1

119909119894)

120572minus1

119890minus120573sum119899

119894=1119909120572

119894

times119899

prod119894=1

[120573 + (1 + 2120573119909120572119894) 119890minus120573119909

120572

119894 ]

(24)

The log-likelihood function is

L (120572 120573 x) = 119899 ln120572 + 119899 ln120573 minus 119899 ln (1 + 120573)

+ (120572 minus 1)119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894+119899

sum119894=1

ln119860119894(120572 120573)

(25)

where

119860119894(120572 120573) = 120573 + (1 + 2120573119909120572

119894) 119890minus120573119909

120572

119894 119894 = 1 119899 (26)

Thefirst partial derivatives of L with respect to 120572 and 120573 are

L120572=119899

120572+119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894ln119909119894+119899

sum119894=1

119860119894120572(120572 120573)

119860119894(120572 120573)

L120573=

119899

120573 (1 + 120573)minus119899

sum119894=1

119909120572119894+119899

sum119894=1

119860119894120573(120572 120573)

119860119894(120572 120573)

(27)

where

119860119894120572(120572 120573) =

120597119860119894(120572 120573)

120597120572= 120573119909120572119894(1 minus 2120573119909120572

119894) ln119909119894119890minus120573119909

120572

119894

119860119894120573(120572 120573) =

120597119860119894(120572 120573)

120597120573= 1 + (1 minus 2120573119909120572

119894) 119909120572119894119890minus120573119909

120572

119894

(28)

The second partial derivatives ofL are

L120572120572= minus119899

1205722minus 120573119899

sum119894=1

119909120572119894(ln119909119894)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205722 (120572 120573) minus [119860

119894120572(120572 120573)]

2

[119860119894(120572 120573)]

2

L120572120573= minus119899

sum119894=1

119909120572119894ln119909119894

+119899

sum119894=1

119860119894(120572 120573)119860

119894120572120573(120572 120573) minus 119860

119894120572(120572 120573)119860

119894120573(120572 120573)

[119860119894(120572 120573)]

2

L120573120573= minus119899

1205732+

119899

(1 + 120573)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205732 (120572 120573) minus [119860

119894120573(120572 120573)]

2

[119860119894(120572 120573)]

2

(29)where1198601198941205722 (120572 120573) = 120573119909120572

119894(ln119909119894)2

119890minus120573119909120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

119860119894120572120573(120572 120573) = 119909120572

119894(ln119909119894) 119890minus120573119909

120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

1198601198941205732 (120572 120573) = minus1199092120572

119894119890minus120573119909

120572

119894 (3 minus 2120573119909120572119894)

(30)

The information matrix is

F (120572 120573) = minus [L120572120572

L120572120573

L120572120573

L120573120573

] (31)

The MLE of 120572 and 120573 say and 120573 are the solution of thesystem of nonlinear equations obtained by setting L

120572=

0 and L120573= 0 such that the F( 120573) is positive definite

This system has no analytic solution so numerical methodssuch as the Newton-Raphsonmethod Burden and Faires [5]should be used

Large-Sample Intervals The MLE of the parameters 120572and 120573 are asymptotically normally distributed with meansequal to the true values of 120572 and 120573 and variances given bythe inverse of the information matrix In particular

(

120573) sim 119873

2((120572120573) Fminus1) (32)

where Fminus1 is the inverse of F( 120573) with main diagonalelements F11 and F22 given by

F11 =

F22

F11F22minus F212

F22 =

F11

F11F22minus F212

(33)

Using (32) large-sample (1 minus 120599)100 confidence intervalsfor 120572 and 120573 are

plusmn 1198851205992radicF11 120573 plusmn 119885

1205992radicF22 (34)

where 1198851205992

is the upper 1001205992 quantile of the standardnormal distribution

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Probability and Statistics

0

05

1

15

2

25

3

35

4

Den

sity

0 05 1 15 2 25 3

x

120572 = 11

120572 = 3535

120572 = 5

Figure 2 The TN(120572 120573) density function for some values of 120572 when120573 = 1

5 Characteristics of TN(120572 120573)51 The Hazard Function The hazard rate function ofTN(120572 120573) is

ℎ (119909) =120572120573119909120572minus1 [120573 + (1 + 2120573119909120572) 119890minus120573119909

120572

]

120573 + (1 + 120573119909120572) 119890minus120573119909120572

119909 ge 0 120572 120573 gt 0

(13)

For120572 = 1 the hazard function is unimodal Its limiting valuesat zero and infinity are 120573 and it reaches a maximum value of

(1 + 2119882minus1(119890minus1120573)) 120573

1 +119882minus1(119890minus1120573)

(14)

at

1

120573(1 +119882

minus1(119890minus1

120573)) (15)

where119882minus1(sdot) denotes the Lambert119882 function which is the

inverse of the function 119909119890119909For 120572 = 1 the shape of the hazard function is difficult to

ascertain analyticallyThe shape was determined numericallyby examining the derivative of the hazard out to the 9999thpercentile of the distribution and the results are shown inFigure 3 For 120572 lt 1 the hazard is decreasing except fora small region with 120572 close to 1 and 120573 lt 05 where thehazard is initially decreasing then increasing and finallydecreasing (DID) For 120572 gt 1 the hazard is strictly increasingfor large 120573 (120573 gt 26 in the figure) For smaller 120573 with 120572close to 1 the hazard can be unimodal (for very small 120573) orinitially increasing then decreasing and finally increasing(IDI) (for slightly larger 120573) Figure 4 shows the hazard forfive choices of 120572 and 120573 which demonstrate the five possibleshapes

00 05 10 15 20 25 30 35

120572

n n n n00

05

10

15

20

25

30

35

120573

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminusminus

minus

minusminusminusminusminusminusminusminusminus

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

+++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++

+

++++++

++++++++++++++++++++++++++

++++++++

++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

++++++++++

+++++++++++++++++

+++++++

++++++++

+++++++++++++++++++

+++++

+

++

+++++++++

++++++++++++++++++

++++++

+++++++++++++++++++++++++++

++++++

+++++++++++++++++++++++++++

+++++

++++++++++++++++++++++++++++

+++

+++++++++++++++

++

+

++

++++++

+

+++++++++

++

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

n

nn

iiiiiiiiiiiiiiiiii

iiiiiiiiiii

i

iiiiiiiii ii ii ii i i i i

iiiiiddd

d

d

DecreasingIncreasingUnimodal

IDIDID

Figure 3 Shapes of the hazard function of TN(120572 120573)

52 Quantiles and Random Sample Generations The 100119902thquantile of TN(120572 120573) 119909

119902 can be derived from that of119873(120573) 119905

119902

as follows

119909119902= 1199051120572119902 (16)

Figure 5 depicts the three quartiles11987611198762 and119876

3 which can

be obtained from the 119902th quantile by setting 119902 = 025 050and 075 respectively From Figure 5 the Interquartile range(IQR = 119876

3minus 1198761) decreases dramatically when 120572 increases

The following algorithm generates a random variate fromTN(120572 120573)

Algorithm 4 (1) Generate 119879 from 119873(120573) using one of theAlgorithms 1ndash3

(2) Set119883 = 1198791120572

53 The Moments and Shape Measures Let 119883 followTN(120572 120573) After some algebra the 119896th ordinary moment of119883is derived as

120583119896= 119864 (119883119896)

= 1198871

Γ (119896120572 + 1)

120573119896120572+ 1198871

Γ (119896120572 + 1)

(2120573)119896120572

+ 1198873

Γ (119896120572 + 2)

(2120573)119896120572

=119896Γ (119896120572) [2120572 (1 + 2119896120572120573) + 119896]

21205722 (1 + 120573) (2120573)119896120572

(17)

Journal of Probability and Statistics 5

Therefore the mean and variance of119883 are

120583 =Γ (1120572) [2120572 (1 + 21120572120573) + 1]

21205722 (1 + 120573) (2120573)1120572

1205902 = (81205722 (1 + 120573) Γ (2

120572) [120572 (1 + 22120572120573) + 1] minus Γ2 (

1

120572)

times[2120572 (1 + 21120572120573) + 1]2

)

times (41205724(1 + 120573)2

(2120573)2120572

)minus1

(18)

Figure 6 depicts the mean and variance of TN(120572 120573) asfunctions of 120572 when 120573 = 1 which shows that the meandecreases dramatically in 120572 and takes its minimum of 08298at 120572 = 161 then it increases steadily to take its maximum of09687 while the variance is decreasing

Based on the first four ordinary moments the measuresof skewness (sk) and kurtosis (120581) of TN(120572 120573) can be obtainedby substituting (17) into (9) and (10) respectively Plots of theskewness and kurtosis of TN(120572 120573) distribution as functionsof 120572 when 120573 = 10 are given in Figure 7 From these plots(1) the skewness is positive when 120572 lt 3535 and negativewhen 120572 gt 3535 and the kurtosis is (i) equal to 3 when either120572 = 24977 or 120572 = 49151 which means that the distributionis mesokurtic (ii) greater than 3 when either 120572 lt 24977 or120572 gt 49151 which means that the distribution is leptokurtic(iii) smaller than 3 when 24977 lt 120572 lt 49151 which meansthat the distribution is platykurtic This analysis shows howthe power parameter 120572 improves 119873(120573) because the powertransformationmodel can be used for datawith awide varietyof distributional shapes

54 Order Statistics Consider 119899 independent and identi-cal components whose lifetimes say 119883

1 1198832 119883

119899 fol-

low TN(120572 120573) The following theorem gives the limit-ing distributions of the lifetime of the series system1198831119899= min119883

1 119883

119899 and of the parallel system 119883

119899119899=

max1198831 119883

119899 consisting of these 119899 components

Theorem 5 The limiting distributions of1198831119899

and119883119899119899

are

lim119899rarrinfin

1198751198831119899minus 119886119899

119887119899

le 119909 = 1 minus 119890minus119909120572

119909 gt 0 (19)

lim119899rarrinfin

119875119883119899119899minus 119888119899

119889119899

le 119909 = exp minus exp (minus119909)

119909 isin (minusinfininfin)

(20)

where 119886119899= 0 119887119899= 119865minus1(1119899) 119888

119899= 119865minus1(1 minus 1119899) 119889

119899= 1119899119891(119888

119899)

and 119865(119909) = 1 minus 119878(119909) is the cdf of 119883

Proof Using LrsquoHospital rule

lim119905rarr0+

119865 (119905119909)

119865 (119905)= lim119905rarr0+

119909119891 (119905119909)

119891 (119905)

= lim119905rarr0+

119909120572 [120573 + (1 + 2120573(119905119909)120572) 119890minus120573(119905119909)120572

] 119890minus120573(119905119909)120572

[120573 + (1 + 2120573119905120572) 119890minus120573119905120572

] 119890minus120573119905120572

= 119909120572

(21)

Therefore (19) follows by Theorem 836(ii) of Arnold et al[4]

For the power transformation we have

lim119909rarrinfin

119889

119889119909

1

ℎ (119909)

= lim119909rarrinfin

minus(120572 minus 1) (120573 + (1 + 120573119909120572) 119890minus120573119909

120572

)

120572120573119909120572 (120573 + (1 + 2 120573119909120572) 119890minus120573119909120572

)

+ lim119909rarrinfin

120573119909120572

1 + 2120573119909120572 + 120573119890120573119909120572

+ lim119909rarrinfin

120573 (2120573119909120572 minus 1) + (120573119909120572 (1 + 2120573119909120572) minus 1) 119890minus120573119909120572

[120573 + (1 + 2120573119909120572) 119890minus120573119909120572

]2

119890120573119909120572

= 0

(22)

Therefore (20) follows by Theorem 833 of Arnold et al [4]

The following Theorem gives the limiting distribution ofthe 119894th order statistic of the 119899 lifetimes119883

1 1198832 119883

119899

Theorem 6 The limiting distributions of119883119894119899 1 le 119894 le 119899 are

lim119899rarrinfin

119875119883119894119899minus 119886119899

119887119899

le 119909 = 1 minus119894minus1

sum119895=0

119909119895120572

119895119890minus119909120572

119909 gt 0 (23)

where 119886119899= 0 119887119899= 119865minus1(1119899)

Proof It follows fromTheorem 5 and (842) of Arnold et al[4]

Theorem 5means that (1198831119899minus119886119899)119887119899and (119883

119899119899minus119888119899)119889119899fol-

low asymptotically119882(1 120572) and Ext(0 1) respectively whileTheorem 6 means that (119883

119894119899minus 119886119899)119887119899follows asymptotically

PG(120572 119894 1)

6 Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is one of the mostcommon methods for estimating the parameters of a sta-tistical model Assume that 119899 independent and identicalitems whose lifetimes follow TN(120572 120573) are put on a lifetest simultaneously Let 119909

1 1199092 119909

119899be the failure times of

6 Journal of Probability and Statistics

0 10 20 30 40

00

05

10

15

x

Haz

ard

(a) 120572 = 05 120573 = 01

x

Haz

ard

0 20 40 60 80 100

006

010

014

018

(b) 120572 = 09 120573 = 01

Haz

ard

x

0 20 40 60 80 100

010

012

014

(c) 120572 = 1 120573 = 01

Haz

ard

x

0 2 4 6 8 10

02

04

06

08

10

(d) 120572 = 12 120573 = 05

Haz

ard

x

0 1 2 3 4 5

00

10

20

30

(e) 120572 = 25 120573 = 01

Figure 4 The hazard function of TN(120572 120573) for some parameter values

Journal of Probability and Statistics 7

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

120572

Q1

Q2

Q3

Figure 5 The quartiles of TN(120572 120573) as functions of 120572 when 120573 = 1

the items and let x = (1199091 119909

119899)The likelihood function for

(120572 120573) is

119871 (120572 120573 x) = 120572119899120573119899

(1 + 120573)119899(119899

prod119894=1

119909119894)

120572minus1

119890minus120573sum119899

119894=1119909120572

119894

times119899

prod119894=1

[120573 + (1 + 2120573119909120572119894) 119890minus120573119909

120572

119894 ]

(24)

The log-likelihood function is

L (120572 120573 x) = 119899 ln120572 + 119899 ln120573 minus 119899 ln (1 + 120573)

+ (120572 minus 1)119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894+119899

sum119894=1

ln119860119894(120572 120573)

(25)

where

119860119894(120572 120573) = 120573 + (1 + 2120573119909120572

119894) 119890minus120573119909

120572

119894 119894 = 1 119899 (26)

Thefirst partial derivatives of L with respect to 120572 and 120573 are

L120572=119899

120572+119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894ln119909119894+119899

sum119894=1

119860119894120572(120572 120573)

119860119894(120572 120573)

L120573=

119899

120573 (1 + 120573)minus119899

sum119894=1

119909120572119894+119899

sum119894=1

119860119894120573(120572 120573)

119860119894(120572 120573)

(27)

where

119860119894120572(120572 120573) =

120597119860119894(120572 120573)

120597120572= 120573119909120572119894(1 minus 2120573119909120572

119894) ln119909119894119890minus120573119909

120572

119894

119860119894120573(120572 120573) =

120597119860119894(120572 120573)

120597120573= 1 + (1 minus 2120573119909120572

119894) 119909120572119894119890minus120573119909

120572

119894

(28)

The second partial derivatives ofL are

L120572120572= minus119899

1205722minus 120573119899

sum119894=1

119909120572119894(ln119909119894)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205722 (120572 120573) minus [119860

119894120572(120572 120573)]

2

[119860119894(120572 120573)]

2

L120572120573= minus119899

sum119894=1

119909120572119894ln119909119894

+119899

sum119894=1

119860119894(120572 120573)119860

119894120572120573(120572 120573) minus 119860

119894120572(120572 120573)119860

119894120573(120572 120573)

[119860119894(120572 120573)]

2

L120573120573= minus119899

1205732+

119899

(1 + 120573)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205732 (120572 120573) minus [119860

119894120573(120572 120573)]

2

[119860119894(120572 120573)]

2

(29)where1198601198941205722 (120572 120573) = 120573119909120572

119894(ln119909119894)2

119890minus120573119909120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

119860119894120572120573(120572 120573) = 119909120572

119894(ln119909119894) 119890minus120573119909

120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

1198601198941205732 (120572 120573) = minus1199092120572

119894119890minus120573119909

120572

119894 (3 minus 2120573119909120572119894)

(30)

The information matrix is

F (120572 120573) = minus [L120572120572

L120572120573

L120572120573

L120573120573

] (31)

The MLE of 120572 and 120573 say and 120573 are the solution of thesystem of nonlinear equations obtained by setting L

120572=

0 and L120573= 0 such that the F( 120573) is positive definite

This system has no analytic solution so numerical methodssuch as the Newton-Raphsonmethod Burden and Faires [5]should be used

Large-Sample Intervals The MLE of the parameters 120572and 120573 are asymptotically normally distributed with meansequal to the true values of 120572 and 120573 and variances given bythe inverse of the information matrix In particular

(

120573) sim 119873

2((120572120573) Fminus1) (32)

where Fminus1 is the inverse of F( 120573) with main diagonalelements F11 and F22 given by

F11 =

F22

F11F22minus F212

F22 =

F11

F11F22minus F212

(33)

Using (32) large-sample (1 minus 120599)100 confidence intervalsfor 120572 and 120573 are

plusmn 1198851205992radicF11 120573 plusmn 119885

1205992radicF22 (34)

where 1198851205992

is the upper 1001205992 quantile of the standardnormal distribution

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 5

Therefore the mean and variance of119883 are

120583 =Γ (1120572) [2120572 (1 + 21120572120573) + 1]

21205722 (1 + 120573) (2120573)1120572

1205902 = (81205722 (1 + 120573) Γ (2

120572) [120572 (1 + 22120572120573) + 1] minus Γ2 (

1

120572)

times[2120572 (1 + 21120572120573) + 1]2

)

times (41205724(1 + 120573)2

(2120573)2120572

)minus1

(18)

Figure 6 depicts the mean and variance of TN(120572 120573) asfunctions of 120572 when 120573 = 1 which shows that the meandecreases dramatically in 120572 and takes its minimum of 08298at 120572 = 161 then it increases steadily to take its maximum of09687 while the variance is decreasing

Based on the first four ordinary moments the measuresof skewness (sk) and kurtosis (120581) of TN(120572 120573) can be obtainedby substituting (17) into (9) and (10) respectively Plots of theskewness and kurtosis of TN(120572 120573) distribution as functionsof 120572 when 120573 = 10 are given in Figure 7 From these plots(1) the skewness is positive when 120572 lt 3535 and negativewhen 120572 gt 3535 and the kurtosis is (i) equal to 3 when either120572 = 24977 or 120572 = 49151 which means that the distributionis mesokurtic (ii) greater than 3 when either 120572 lt 24977 or120572 gt 49151 which means that the distribution is leptokurtic(iii) smaller than 3 when 24977 lt 120572 lt 49151 which meansthat the distribution is platykurtic This analysis shows howthe power parameter 120572 improves 119873(120573) because the powertransformationmodel can be used for datawith awide varietyof distributional shapes

54 Order Statistics Consider 119899 independent and identi-cal components whose lifetimes say 119883

1 1198832 119883

119899 fol-

low TN(120572 120573) The following theorem gives the limit-ing distributions of the lifetime of the series system1198831119899= min119883

1 119883

119899 and of the parallel system 119883

119899119899=

max1198831 119883

119899 consisting of these 119899 components

Theorem 5 The limiting distributions of1198831119899

and119883119899119899

are

lim119899rarrinfin

1198751198831119899minus 119886119899

119887119899

le 119909 = 1 minus 119890minus119909120572

119909 gt 0 (19)

lim119899rarrinfin

119875119883119899119899minus 119888119899

119889119899

le 119909 = exp minus exp (minus119909)

119909 isin (minusinfininfin)

(20)

where 119886119899= 0 119887119899= 119865minus1(1119899) 119888

119899= 119865minus1(1 minus 1119899) 119889

119899= 1119899119891(119888

119899)

and 119865(119909) = 1 minus 119878(119909) is the cdf of 119883

Proof Using LrsquoHospital rule

lim119905rarr0+

119865 (119905119909)

119865 (119905)= lim119905rarr0+

119909119891 (119905119909)

119891 (119905)

= lim119905rarr0+

119909120572 [120573 + (1 + 2120573(119905119909)120572) 119890minus120573(119905119909)120572

] 119890minus120573(119905119909)120572

[120573 + (1 + 2120573119905120572) 119890minus120573119905120572

] 119890minus120573119905120572

= 119909120572

(21)

Therefore (19) follows by Theorem 836(ii) of Arnold et al[4]

For the power transformation we have

lim119909rarrinfin

119889

119889119909

1

ℎ (119909)

= lim119909rarrinfin

minus(120572 minus 1) (120573 + (1 + 120573119909120572) 119890minus120573119909

120572

)

120572120573119909120572 (120573 + (1 + 2 120573119909120572) 119890minus120573119909120572

)

+ lim119909rarrinfin

120573119909120572

1 + 2120573119909120572 + 120573119890120573119909120572

+ lim119909rarrinfin

120573 (2120573119909120572 minus 1) + (120573119909120572 (1 + 2120573119909120572) minus 1) 119890minus120573119909120572

[120573 + (1 + 2120573119909120572) 119890minus120573119909120572

]2

119890120573119909120572

= 0

(22)

Therefore (20) follows by Theorem 833 of Arnold et al [4]

The following Theorem gives the limiting distribution ofthe 119894th order statistic of the 119899 lifetimes119883

1 1198832 119883

119899

Theorem 6 The limiting distributions of119883119894119899 1 le 119894 le 119899 are

lim119899rarrinfin

119875119883119894119899minus 119886119899

119887119899

le 119909 = 1 minus119894minus1

sum119895=0

119909119895120572

119895119890minus119909120572

119909 gt 0 (23)

where 119886119899= 0 119887119899= 119865minus1(1119899)

Proof It follows fromTheorem 5 and (842) of Arnold et al[4]

Theorem 5means that (1198831119899minus119886119899)119887119899and (119883

119899119899minus119888119899)119889119899fol-

low asymptotically119882(1 120572) and Ext(0 1) respectively whileTheorem 6 means that (119883

119894119899minus 119886119899)119887119899follows asymptotically

PG(120572 119894 1)

6 Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is one of the mostcommon methods for estimating the parameters of a sta-tistical model Assume that 119899 independent and identicalitems whose lifetimes follow TN(120572 120573) are put on a lifetest simultaneously Let 119909

1 1199092 119909

119899be the failure times of

6 Journal of Probability and Statistics

0 10 20 30 40

00

05

10

15

x

Haz

ard

(a) 120572 = 05 120573 = 01

x

Haz

ard

0 20 40 60 80 100

006

010

014

018

(b) 120572 = 09 120573 = 01

Haz

ard

x

0 20 40 60 80 100

010

012

014

(c) 120572 = 1 120573 = 01

Haz

ard

x

0 2 4 6 8 10

02

04

06

08

10

(d) 120572 = 12 120573 = 05

Haz

ard

x

0 1 2 3 4 5

00

10

20

30

(e) 120572 = 25 120573 = 01

Figure 4 The hazard function of TN(120572 120573) for some parameter values

Journal of Probability and Statistics 7

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

120572

Q1

Q2

Q3

Figure 5 The quartiles of TN(120572 120573) as functions of 120572 when 120573 = 1

the items and let x = (1199091 119909

119899)The likelihood function for

(120572 120573) is

119871 (120572 120573 x) = 120572119899120573119899

(1 + 120573)119899(119899

prod119894=1

119909119894)

120572minus1

119890minus120573sum119899

119894=1119909120572

119894

times119899

prod119894=1

[120573 + (1 + 2120573119909120572119894) 119890minus120573119909

120572

119894 ]

(24)

The log-likelihood function is

L (120572 120573 x) = 119899 ln120572 + 119899 ln120573 minus 119899 ln (1 + 120573)

+ (120572 minus 1)119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894+119899

sum119894=1

ln119860119894(120572 120573)

(25)

where

119860119894(120572 120573) = 120573 + (1 + 2120573119909120572

119894) 119890minus120573119909

120572

119894 119894 = 1 119899 (26)

Thefirst partial derivatives of L with respect to 120572 and 120573 are

L120572=119899

120572+119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894ln119909119894+119899

sum119894=1

119860119894120572(120572 120573)

119860119894(120572 120573)

L120573=

119899

120573 (1 + 120573)minus119899

sum119894=1

119909120572119894+119899

sum119894=1

119860119894120573(120572 120573)

119860119894(120572 120573)

(27)

where

119860119894120572(120572 120573) =

120597119860119894(120572 120573)

120597120572= 120573119909120572119894(1 minus 2120573119909120572

119894) ln119909119894119890minus120573119909

120572

119894

119860119894120573(120572 120573) =

120597119860119894(120572 120573)

120597120573= 1 + (1 minus 2120573119909120572

119894) 119909120572119894119890minus120573119909

120572

119894

(28)

The second partial derivatives ofL are

L120572120572= minus119899

1205722minus 120573119899

sum119894=1

119909120572119894(ln119909119894)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205722 (120572 120573) minus [119860

119894120572(120572 120573)]

2

[119860119894(120572 120573)]

2

L120572120573= minus119899

sum119894=1

119909120572119894ln119909119894

+119899

sum119894=1

119860119894(120572 120573)119860

119894120572120573(120572 120573) minus 119860

119894120572(120572 120573)119860

119894120573(120572 120573)

[119860119894(120572 120573)]

2

L120573120573= minus119899

1205732+

119899

(1 + 120573)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205732 (120572 120573) minus [119860

119894120573(120572 120573)]

2

[119860119894(120572 120573)]

2

(29)where1198601198941205722 (120572 120573) = 120573119909120572

119894(ln119909119894)2

119890minus120573119909120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

119860119894120572120573(120572 120573) = 119909120572

119894(ln119909119894) 119890minus120573119909

120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

1198601198941205732 (120572 120573) = minus1199092120572

119894119890minus120573119909

120572

119894 (3 minus 2120573119909120572119894)

(30)

The information matrix is

F (120572 120573) = minus [L120572120572

L120572120573

L120572120573

L120573120573

] (31)

The MLE of 120572 and 120573 say and 120573 are the solution of thesystem of nonlinear equations obtained by setting L

120572=

0 and L120573= 0 such that the F( 120573) is positive definite

This system has no analytic solution so numerical methodssuch as the Newton-Raphsonmethod Burden and Faires [5]should be used

Large-Sample Intervals The MLE of the parameters 120572and 120573 are asymptotically normally distributed with meansequal to the true values of 120572 and 120573 and variances given bythe inverse of the information matrix In particular

(

120573) sim 119873

2((120572120573) Fminus1) (32)

where Fminus1 is the inverse of F( 120573) with main diagonalelements F11 and F22 given by

F11 =

F22

F11F22minus F212

F22 =

F11

F11F22minus F212

(33)

Using (32) large-sample (1 minus 120599)100 confidence intervalsfor 120572 and 120573 are

plusmn 1198851205992radicF11 120573 plusmn 119885

1205992radicF22 (34)

where 1198851205992

is the upper 1001205992 quantile of the standardnormal distribution

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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Stochastic AnalysisInternational Journal of

6 Journal of Probability and Statistics

0 10 20 30 40

00

05

10

15

x

Haz

ard

(a) 120572 = 05 120573 = 01

x

Haz

ard

0 20 40 60 80 100

006

010

014

018

(b) 120572 = 09 120573 = 01

Haz

ard

x

0 20 40 60 80 100

010

012

014

(c) 120572 = 1 120573 = 01

Haz

ard

x

0 2 4 6 8 10

02

04

06

08

10

(d) 120572 = 12 120573 = 05

Haz

ard

x

0 1 2 3 4 5

00

10

20

30

(e) 120572 = 25 120573 = 01

Figure 4 The hazard function of TN(120572 120573) for some parameter values

Journal of Probability and Statistics 7

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

120572

Q1

Q2

Q3

Figure 5 The quartiles of TN(120572 120573) as functions of 120572 when 120573 = 1

the items and let x = (1199091 119909

119899)The likelihood function for

(120572 120573) is

119871 (120572 120573 x) = 120572119899120573119899

(1 + 120573)119899(119899

prod119894=1

119909119894)

120572minus1

119890minus120573sum119899

119894=1119909120572

119894

times119899

prod119894=1

[120573 + (1 + 2120573119909120572119894) 119890minus120573119909

120572

119894 ]

(24)

The log-likelihood function is

L (120572 120573 x) = 119899 ln120572 + 119899 ln120573 minus 119899 ln (1 + 120573)

+ (120572 minus 1)119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894+119899

sum119894=1

ln119860119894(120572 120573)

(25)

where

119860119894(120572 120573) = 120573 + (1 + 2120573119909120572

119894) 119890minus120573119909

120572

119894 119894 = 1 119899 (26)

Thefirst partial derivatives of L with respect to 120572 and 120573 are

L120572=119899

120572+119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894ln119909119894+119899

sum119894=1

119860119894120572(120572 120573)

119860119894(120572 120573)

L120573=

119899

120573 (1 + 120573)minus119899

sum119894=1

119909120572119894+119899

sum119894=1

119860119894120573(120572 120573)

119860119894(120572 120573)

(27)

where

119860119894120572(120572 120573) =

120597119860119894(120572 120573)

120597120572= 120573119909120572119894(1 minus 2120573119909120572

119894) ln119909119894119890minus120573119909

120572

119894

119860119894120573(120572 120573) =

120597119860119894(120572 120573)

120597120573= 1 + (1 minus 2120573119909120572

119894) 119909120572119894119890minus120573119909

120572

119894

(28)

The second partial derivatives ofL are

L120572120572= minus119899

1205722minus 120573119899

sum119894=1

119909120572119894(ln119909119894)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205722 (120572 120573) minus [119860

119894120572(120572 120573)]

2

[119860119894(120572 120573)]

2

L120572120573= minus119899

sum119894=1

119909120572119894ln119909119894

+119899

sum119894=1

119860119894(120572 120573)119860

119894120572120573(120572 120573) minus 119860

119894120572(120572 120573)119860

119894120573(120572 120573)

[119860119894(120572 120573)]

2

L120573120573= minus119899

1205732+

119899

(1 + 120573)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205732 (120572 120573) minus [119860

119894120573(120572 120573)]

2

[119860119894(120572 120573)]

2

(29)where1198601198941205722 (120572 120573) = 120573119909120572

119894(ln119909119894)2

119890minus120573119909120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

119860119894120572120573(120572 120573) = 119909120572

119894(ln119909119894) 119890minus120573119909

120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

1198601198941205732 (120572 120573) = minus1199092120572

119894119890minus120573119909

120572

119894 (3 minus 2120573119909120572119894)

(30)

The information matrix is

F (120572 120573) = minus [L120572120572

L120572120573

L120572120573

L120573120573

] (31)

The MLE of 120572 and 120573 say and 120573 are the solution of thesystem of nonlinear equations obtained by setting L

120572=

0 and L120573= 0 such that the F( 120573) is positive definite

This system has no analytic solution so numerical methodssuch as the Newton-Raphsonmethod Burden and Faires [5]should be used

Large-Sample Intervals The MLE of the parameters 120572and 120573 are asymptotically normally distributed with meansequal to the true values of 120572 and 120573 and variances given bythe inverse of the information matrix In particular

(

120573) sim 119873

2((120572120573) Fminus1) (32)

where Fminus1 is the inverse of F( 120573) with main diagonalelements F11 and F22 given by

F11 =

F22

F11F22minus F212

F22 =

F11

F11F22minus F212

(33)

Using (32) large-sample (1 minus 120599)100 confidence intervalsfor 120572 and 120573 are

plusmn 1198851205992radicF11 120573 plusmn 119885

1205992radicF22 (34)

where 1198851205992

is the upper 1001205992 quantile of the standardnormal distribution

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 7

0 1 2 3 4 5 6 7 8 9 100

05

1

15

2

25

3

120572

Q1

Q2

Q3

Figure 5 The quartiles of TN(120572 120573) as functions of 120572 when 120573 = 1

the items and let x = (1199091 119909

119899)The likelihood function for

(120572 120573) is

119871 (120572 120573 x) = 120572119899120573119899

(1 + 120573)119899(119899

prod119894=1

119909119894)

120572minus1

119890minus120573sum119899

119894=1119909120572

119894

times119899

prod119894=1

[120573 + (1 + 2120573119909120572119894) 119890minus120573119909

120572

119894 ]

(24)

The log-likelihood function is

L (120572 120573 x) = 119899 ln120572 + 119899 ln120573 minus 119899 ln (1 + 120573)

+ (120572 minus 1)119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894+119899

sum119894=1

ln119860119894(120572 120573)

(25)

where

119860119894(120572 120573) = 120573 + (1 + 2120573119909120572

119894) 119890minus120573119909

120572

119894 119894 = 1 119899 (26)

Thefirst partial derivatives of L with respect to 120572 and 120573 are

L120572=119899

120572+119899

sum119894=1

ln119909119894minus 120573119899

sum119894=1

119909120572119894ln119909119894+119899

sum119894=1

119860119894120572(120572 120573)

119860119894(120572 120573)

L120573=

119899

120573 (1 + 120573)minus119899

sum119894=1

119909120572119894+119899

sum119894=1

119860119894120573(120572 120573)

119860119894(120572 120573)

(27)

where

119860119894120572(120572 120573) =

120597119860119894(120572 120573)

120597120572= 120573119909120572119894(1 minus 2120573119909120572

119894) ln119909119894119890minus120573119909

120572

119894

119860119894120573(120572 120573) =

120597119860119894(120572 120573)

120597120573= 1 + (1 minus 2120573119909120572

119894) 119909120572119894119890minus120573119909

120572

119894

(28)

The second partial derivatives ofL are

L120572120572= minus119899

1205722minus 120573119899

sum119894=1

119909120572119894(ln119909119894)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205722 (120572 120573) minus [119860

119894120572(120572 120573)]

2

[119860119894(120572 120573)]

2

L120572120573= minus119899

sum119894=1

119909120572119894ln119909119894

+119899

sum119894=1

119860119894(120572 120573)119860

119894120572120573(120572 120573) minus 119860

119894120572(120572 120573)119860

119894120573(120572 120573)

[119860119894(120572 120573)]

2

L120573120573= minus119899

1205732+

119899

(1 + 120573)2

+119899

sum119894=1

119860119894(120572 120573)119860

1198941205732 (120572 120573) minus [119860

119894120573(120572 120573)]

2

[119860119894(120572 120573)]

2

(29)where1198601198941205722 (120572 120573) = 120573119909120572

119894(ln119909119894)2

119890minus120573119909120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

119860119894120572120573(120572 120573) = 119909120572

119894(ln119909119894) 119890minus120573119909

120572

119894 (1 minus 5120573119909120572119894+ 212057321199092120572

119894)

1198601198941205732 (120572 120573) = minus1199092120572

119894119890minus120573119909

120572

119894 (3 minus 2120573119909120572119894)

(30)

The information matrix is

F (120572 120573) = minus [L120572120572

L120572120573

L120572120573

L120573120573

] (31)

The MLE of 120572 and 120573 say and 120573 are the solution of thesystem of nonlinear equations obtained by setting L

120572=

0 and L120573= 0 such that the F( 120573) is positive definite

This system has no analytic solution so numerical methodssuch as the Newton-Raphsonmethod Burden and Faires [5]should be used

Large-Sample Intervals The MLE of the parameters 120572and 120573 are asymptotically normally distributed with meansequal to the true values of 120572 and 120573 and variances given bythe inverse of the information matrix In particular

(

120573) sim 119873

2((120572120573) Fminus1) (32)

where Fminus1 is the inverse of F( 120573) with main diagonalelements F11 and F22 given by

F11 =

F22

F11F22minus F212

F22 =

F11

F11F22minus F212

(33)

Using (32) large-sample (1 minus 120599)100 confidence intervalsfor 120572 and 120573 are

plusmn 1198851205992radicF11 120573 plusmn 119885

1205992radicF22 (34)

where 1198851205992

is the upper 1001205992 quantile of the standardnormal distribution

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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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

8 Journal of Probability and Statistics

0 5 10 15 2008

09

11

12

13

14

15M

ean

1205720 5 10 15 20

120572

1

0

05

1

15

2

25

3

Varia

nce

120583 = 09687

120583 = 08298

Figure 6 The mean and variance of TN(120572 120573) as functions of 120572 when 120573 = 1

0 5 10 2015 0 5 10 2015minus2

minus1

0

1

2

3

4

5

6

0

1

2

3

4

5

6

Skew

ness

Kurt

osis

120572 120572

120572 = 35318

(120572 = 24977

(120572 = 49151

(120572 = 33733 = 28337)

120581 =

120581

3

120581 =

)

3)

Figure 7 The skewness and kurtosis of TN(120572 120573) as functions of 120572 when 120573 = 1

7 Applications

In this section we analyze four data sets to illustrate theapplicability of the two new distributions proposed in thispaper The first data consists of 61 observed recidivismfailure times (in days) of individuals released directly fromcorrectional institutions to parole in theDistrict of ColumbiaColumbia USA [6] The second data set consists of 43active repair times (in hours) for an airborne communicationtransceiver [7] The third data set consists of 57 times (in

thousands of operating hours) of unscheduled maintenanceactions for the number 4 diesel engine of the USS Grampusup to 16 thousand hours of operation [8] The forth dataset consists of the tensile strength (measured in GPa) of 69carbon fibers tested under tension at gauge lengths of 20mm[9]

We will refer to these data sets as failure times repairtimes maintenance actions and tensile strength data respec-tively For each data set we fit the proposed two distributionsas well as Lindley and power Lindley distributions For

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 9

0

02

04

06

08

0 02 04 06 08

1

1

u

0 100 200 300 400 5000

0002

0004

0006

0008

001

x

0 100 200 300 400 500

x

0 100 200 300 400 500

x

Haz

ard

0

1

2

3

times10minus3

4

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

N(120573)

TN(120572 120573)

TN(120572 120573)

L(120573)

L(120573)

PL(120572 120573)

PL(120572 120573)

Figure 8 The TTT-transform fitted hazard pdf and cdf of the failure times data

the sake of comparison we apply goodness-of-fit tests toverify which distribution better fits these data sets Weconsider the well-known Kolmogorov-Smirnov (K-S) statis-tic the Cramer-von Mises (C-M) and Anderson-Darling(A-D) statistics [10] Furthermore we consider the Akaikeinformation criterion AIC = minus2L + 2119902 where L is the log-likelihood function at theMLE of the parameters and 119902 is thenumber of model parameters Table 1 shows the MLE of theparameters of each model the corresponding maximum log-likelihood value and the AIC for the four data sets Table 2presents the results of the goodness of fit tests for the fourdata sets using each model

For every data set we plotted (1) the scaled total time ontest transform (TTT-transform) plot which gives qualitativeinformation about the hazard rate shape [11] (2) the hazardfunctions for the four fitted models (3) the empirical andfitted density and distribution functions Figures 8 9 10 and11 show the four plots for the four data sets 1ndash4 respectivelyThe scaled TTT-transform plots show that the repair data

set has a unimodal hazard while the rest of data sets haveincreasing hazards

The inverse of information matrix at the MLE using thefour data sets are listed below

Failure times

Fminus1 = [

1717231 times 10minus3 minus1024661 times 10minus5

minus1024661 times 10minus5 5019728 times 10minus7] (35)

Active repair

Fminus1 = [

000901512 minus000442645minus000442645 000430030

] (36)

Maintenance actions

Fminus1 = [

0046064730 minus1998013 times 10minus3

minus1998013 times 10minus3 907815 times 10minus5] (37)

Tensile strength

Fminus1 = [

0521162636 minus3846724 times 10minus3

minus3846724 times 10minus3 2892066 times 10minus5] (38)

10 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

1

u

0 5 10 15 20 250

01

02

03

04

05

x

0 5 10 15 20 25

x

Haz

ard

0 2 4 6 8 10 120

005

01

015

02

025

x

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 9 The TTT-transform hazard pdf and cdf of the active repair times data

Table 1 Parameter estimates maximum log-likelihood and AIC for the four data sets

Data Model Parameter estimates L AIC

Failure times

L(120573) 120573 = 9403 times 10minus3 minus386894 775789PL(120572 120573) = 08778 120573 = 00185 minus385958 775916N(120573) 120573 = 361354 times 10minus3 minus384872 771743TN(120572 120573) = 12551 120573 = 8442 times 10minus4 minus382904 769808

Repair times

L(120573) 120573 = 0449891 minus104433 210865PL(120572 120573) = 0762291 120573 = 0652788 minus99974 203947N(120573) 120573 = 0226013 minus100821 203642TN(120572 120573) = 0843574 120573 = 0295802 minus99572 203144

Maintenance actions

L(120573) 120573 = 0217325 minus170317 342634PL(120572 120573) = 128229 120573 = 0117289 minus167089 338178N(120573) 120573 = 00936263 minus174636 351271TN(120572 120573) = 169539 120573 = 00172624 minus165584 335167

Tensile strength

L(120573) 120573 = 06545 minus119190 240381PL(120572 120573) = 38678 120573 = 00497 minus49060 102119N(120573) 120573 = 03334 minus124618 251235TN(120572 120573) = 48853 120573 = 00067 minus49976 103952

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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

Journal of Probability and Statistics 11

0

02

04

06

08

0 02 04 06 08

1

1

u x

Haz

ard

0 5 10 15

x

0 5 10 15

x

0 5 10 15

0

005

01

015

02

025

03

035

0

002

004

006

008

01

Den

sity

0

02

04

06

08

1

cdf

Empirical

Scal

ed T

TT-tr

ansfo

rm

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Figure 10 The TTT-transform hazard pdf and cdf of the maintenance actions data

Table 2 Statistics K-S (119875 value) C-M and A-D for the four data sets

Data Model K-S (119875 value) C-M A-D

Failure times

L(120573) 0099087 (055362) 01456121 0907295PL(120572 120573) 0094044 (061929) 01632937 1013002N(120573) 0103937 (049274) 01175828 0741661TN(120572 120573) 0069538 (090959) 00847141 0542494

Repair times

L(120573) 0234632 (001449) 01810547 1210262PL(120572 120573) 0120462 (052160) 01330721 0895534N(120573) 0174114 (013065) 01506954 0993861TN(120572 120573) 0114174 (058940) 01268951 0840214

Maintenance actions

L(120573) 0122002 (033663) 0187688 1305582PL(120572 120573) 0119731 (035863) 0178752 1244465N(120573) 0146367 (015728) 0186369 1298771TN(120572 120573) 0111731 (044316) 0160603 1120745

Tensile strength

L(120573) 0386637 (117899 times 10minus9) 00403019 03025705PL(120572 120573) 0044275 (099840) 00181847 01619866N(120573) 0415150 (457128 times 10minus11) 00432435 03221342TN(120572 120573) 0056805 (096997) 00403920 03132999

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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

12 Journal of Probability and Statistics

0

02

04

06

08

0 02 04 06 08

1

0

02

04

06

08

1

0

02

04

06

08

1

1

u

1 15 2 25 3 35

x

0

1 2 3

x

0

1 2 30

2

4

6

8

x

Haz

ard

Den

sity

cdf

Empirical

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

N(120573)

TN(120572 120573)

L(120573)PL(120572 120573)

Scal

ed T

TT-tr

ansfo

rm

Figure 11 The TTT-transform hazard pdf and cdf of the tensile strength data

Table 3 The LRT and 119875 value for the four data sets

Failuretimes

Repairtimes

Maintenanceactions

Tensilestrength

Λ 3936 2498 18104 149284119875 value 00473 01140 20916 times 10minus5 0

For the first three data sets TN(120572 120573) model has thesmallest value of the Kolmogorov-Smirnov (largest 119875 value)the Cramer-von Mises and Anderson-Darling goodness-of-fit tests statistics which indicate that the best fit is providedby the TN model for these data sets For the forth data setthe power Lindley model provides the best fit in the sense ofhaving the smallest test statistics For all data sets TN(120572 120573)is a better fit than 119873(120573) For the first two data sets 119873(120573) isa better fit than both 119871(120573) and PL(120572 120573) while it is the worst

fit for the last two data sets The AIC statistic is the lowest forTN(120572 120573) for all data sets except for tensile strength where itis slightly higher

Further for testing 119873(120573) as a submodel of the TN(120572 120573)we use the likelihood ratio test statistic (LRT) to check ifthe fit using the TN(120572 120573) is statistically superior to a fitusing the 119873(120573) for each data set The LRT for testing 119867

0

120572 = 1 against 1198671 120572 = 1 is Λ = 2(L

1198671

minus L1198670

) whereL1198671

and L1198670

are the maximum log-likelihood valuesunder 119867

1and 119867

0 respectively Under 119867

0 Λ 119889

997888rarr 12059421

The LRT rejects 1198670if Λ gt 1205942

1120574 where 1205942

1120574denotes the

upper 100120574 point of chi-square distribution with 1 degreeof freedom Table 3 lists the values of the LRT and thecorresponding 119875 value for the four data sets Based on the 119875values the 119873(120573) is not rejected against the TN(120572 120573) to fitthe repair times data set while it is rejected at any level of

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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

Journal of Probability and Statistics 13

Table 4 MSE coverage probability and average width

120572 120573 119899 MSE120572

MSE120573

CP120572

AW120572

CP120573

AW120573

05 1

25 0008889 00639280 09550 0335546 09426 088883550 0003720 00250429 09553 0229186 09541 060807575 0002295 00158072 09558 0184697 09527 0491875100 0001690 00120432 09547 0159216 09528 0425521

10 10

25 0035209 00603732 09552 0669487 09492 088467150 0015100 00259297 09530 0458663 09504 06092275 0009433 00165418 09495 0369965 09516 0493743100 0006956 00121903 09517 0318365 09498 0425539

20 10

25 0144086 00609798 09518 1342590 09435 088372350 0061349 00260667 09472 0916073 09458 060856975 0037822 00163597 09526 0740067 09508 0492714100 0028013 00119553 09492 0638045 09510 0424817

13 05

25 0058995 00173859 09532 0866467 09216 048157450 0024938 00081550 09535 0594661 09348 033948775 0015840 00052273 09557 0480683 09400 0277333100 0011542 00037797 09551 0414303 09451 0239666

05 12

25 0009273 00972282 09480 0335704 09499 108488050 0003832 00381008 09485 0229018 09494 073343175 0002355 00239632 09511 0184765 09565 0593873100 0001787 00173496 09485 0159225 09531 0510691

10 12

25 0036640 00898864 09502 0670473 09556 107518050 0015133 00380503 09525 0457310 09526 073522175 0009416 00240831 09525 0369850 09547 0594754100 0007055 00174628 09486 0318305 09523 0510473

20 12

25 0148128 00929780 09515 1344300 09493 107618050 0061533 00386808 09522 0916556 09522 073467675 0038607 00248240 09495 0738584 09492 0593036100 0028413 00172320 09474 0636959 09541 0509967

13 15

25 0063330 01753970 09534 0875085 09601 141512050 0025851 00651604 09519 0596480 09537 094566675 0016517 00404539 09509 0481512 09536 0759421100 0011529 00292367 09548 0414153 09552 0652792

09 12

25 0029932 00929842 09499 0604276 09495 107417050 0012327 00389142 09501 0411230 09509 073420975 0007675 00238699 09504 0332339 09531 0594194100 0005652 00174031 09531 0286597 09505 0510094

significance greater than or equal to 00473 to fit the otherthree data sets

8 Simulation Study

We used a simulation study to investigate the performanceof the accuracy of point and interval estimates of the twoparameters of TN(120572 120573) The following steps are as follows

(1) Specify the values of the parameters 120572 and 120573

(2) Specify the sample size 119899

(3) Use Algorithm 2 and the transformation to generate arandom sample with size 119899 from TN(120572 120573)

(a) Calculate the MLE of the two parameters andthe inverse of the Fisher matrix

(b) Calculate the squared deviation of the MLEfrom the exact value of each parameter

(c) Calculate a 95 CI for each parameter

(4) Repeat steps 2-3 119873 times(5) Calculate the mean squared error (MSE) the average

of the confidence interval widths and the coverageprobability for each parameter The MSE associatedwith the MLE of the parameter 120599 MSE

120599 is

MSE120599=1

119873

119873

sum119894=1

(120599119894minus 120599)2

(39)

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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

14 Journal of Probability and Statistics

where 120599119894is the MLE of 120599 using the 119894th sample 119894 =

1 2 119873 and 120599 = 120572 120573 Coverage probability is theproportion of the 119873 simulated confidence intervalswhich include the true parameter 120599

The simulation study is used when 119873 = 10 000 the samplesizes are 25 50 75 and 100 and the parameter values (120572 120573) =(05 1) (1 1) (2 1) (13 05) (05 12) (1 12) (2 12)(13 15) and (09 12) Some of the selected valuesof (120572 120573) give decreasing unimodal increasing andincreasing-decreasing-increasing hazard shapes respectivelyas shown in Figure 3 Table 4 presents the MSE coverageprobability (CP

120599) and averagewidth (AW)of 95 confidence

intervals of each parameter As it was expected this tableshows that the MSEs of the estimates decrease as the samplesize increases that the coverage probabilities are very closeto the nominal level of 95 and that the average widthsdecrease as the sample size increases

9 Conclusion

In this paper we have proposed new one-parameter andtwo-parameter distributions called the 119873(120573) and TN(120572 120573)respectively The TN(120572 120573) was obtained by using a powertransformation of the 119873(120573) distributed variable TheTN(120572 120573) provides more flexibility than the 119873(120573) in termsof the shape of the density and hazard rate functions aswell as its skewness and kurtosis We derived the maximumlikelihood estimates of the parameters and their variance-covariance matrix We proposed different algorithms togenerate samples from the two proposed distributionsApplications of the two proposed distributions to realdata sets show better fits than Lindley and power Lindleydistributions Finally we examined the accuracy of themaximum likelihood estimators of the TN(120572 120573) parametersas well as the coverage probability and average width of theconfidence intervals for the parameters using simulation

Notation

pdf Probability density functioncdf Cumulative distribution functionmgf Moment generating function119880(0 1) Uniform distribution on (0 1)Exp(120573) Exponential distribution with mean 1120573119882(] 120573) Weibull distribution with pdf

119891(119909) = ]120573119909]minus1119890minus120573119909]

119866(] 120573) Gamma distribution with pdf119891(119909) = (120573]Γ(]))119909]minus1119890minus120573119909

PG(120572 ] 120573) Power gamma distribution with pdf119891(119909) = (120572120573]Γ(]))119909120572]minus1119890minus120573119909

120572

PL(120572 120573) Power Lindley distribution with pdf119891(119909) = (1205721205732(1 + 120573))119909120572minus1(1 + 119909120572)119890minus120573119909

120572

Ext(120572 120573) Extreme-value distribution with pdf119891(119909) = (1120573)119890minus(119909minus120572)120573minusexpminus(119909minus120572)120573

Conflict of Interests

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

References

[1] D V Lindley ldquoFiducial distributions and Bayesrsquo theoremrdquoJournal of the Royal Statistical Society B Methodological vol 20pp 102ndash107 1958

[2] M E Ghitany B Atieh and S Nadarajah ldquoLindley distributionand its applicationrdquoMathematics and Computers in Simulationvol 78 no 4 pp 493ndash506 2008

[3] M E Ghitany D K Al-Mutairi N Balakrishnan and L J Al-Enezi ldquoPower Lindley distribution and associated inferencerdquoComputational Statistics amp Data Analysis vol 64 pp 20ndash332013

[4] B C Arnold N Balakrishnan and H N Nagaraja A FirstCourse in Order Statistics John Wiley amp Sons New York NYUSA 1992

[5] L Burden and J D Faires Numerical Analysis BrooksColeCengage Learing Stamford Conn USA 9th edition 2011

[6] S Stollmack and C M Harris ldquoFailure-rate analysis applied torecidivism datardquo Operations Research vol 22 no 6 pp 1192ndash1205 1974

[7] B Joslashrgensen Statistical Properties of the Generalized InverseGaussian Distribution vol 9 Springer Heidelberg Germany1982

[8] W OMeeker and L A Escobar Statistical Methods for Reliabil-ity Data John Wiley amp Sons New York NY USA 1998

[9] M G Bader and A M Priest ldquoStatistical aspects of fiber andbundle strength in hybrid compositesrdquo in Progress In Scienceand EngIneerIng Composites T Hayashi K Kawata and SUmekawa Eds pp 1129ndash1136 ICCM-IV Tokyo Japan 1982

[10] G Chen and N Balakrishnan ldquoA general purpose approximategoodnessoffit testrdquo Journal of Quality Technology vol 27 pp154ndash161 1995

[11] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol R-36 no 1 pp 106ndash108 1987

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