Research Article The Complementary Exponentiated ...downloads.hindawi.com/journals/jps/2013/502159.pdf · Journal of Probability and Statistics 0 0.1 0.2 0.3 0.4 0.5 Density Density

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2013 Article ID 502159 12 pageshttpdxdoiorg1011552013502159

Research ArticleThe Complementary Exponentiated Exponential GeometricLifetime Distribution

Francisco Louzada1 Vitor Marchi2 and James Carpenter3

1 Department of Applied Mathematics and Statistics ICMC University of Sao Paulo 13560-970 Sao Carlos SP Brazil2 Department of Statistics Federal University of Sao Paulo 13565-905 Sao Carlos SP Brazil3 London School of Hygiene and Tropical Medicine University of London Keppel Street London WC1E 7HT UK

Correspondence should be addressed to Francisco Louzada louzadaicmcuspbr

Received 8 August 2012 Revised 17 November 2012 Accepted 26 November 2012

Academic Editor Gauss M Cordeiro

Copyright copy 2013 Francisco Louzada 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

We proposed a new family of lifetime distributions namely complementary exponentiated exponential geometric distributionThis new family arises on a latent competing risk scenario where the lifetime associated with a particular risk is not observablebut only the maximum lifetime value among all risks The properties of the proposed distribution are discussed including aformal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions momentsrth moment of the ith order statistic mean residual lifetime and modal value Inference is implemented via a straightforwardlymaximum likelihood procedure The practical importance of the new distribution was demonstrated in three applications whereour distribution outperforms several former lifetime distributions such as the exponential the exponential-geometric theWeibullthe modified Weibull and the generalized exponential-Poisson distribution

1 Introduction

Several new classes of models have been introduced in recentyears grounded in the simple exponential distribution Themain idea is to propose lifetime distributions which canaccommodate practical applications where the underlyinghazard functions are nonconstant presenting monotoneshapes since the exponential distribution does not provide areasonable fit in such situations For instance we can cite [1]which proposed a variation of the exponential distributionthe exponential geometric (EG) distribution with decreasinghazard function [2] which introduced the exponentiatedexponential distribution as a generalization of the usualexponential distribution which can accommodate data withincreasing and decreasing hazard functions [3] which pro-posed a generalized exponential distribution which canaccommodate data with increasing and decreasing hazardfunctions [4] which proposed the exponentiated type distri-butions extending the Frechet gamma Gumbel andWeibulldistributions [5] which proposed another modification ofthe exponential distributionwith decreasing hazard function

[6] which generalizes the distribution proposed by [5] byincluding a power parameter in this distribution which canaccommodate increasing decreasing and unimodal haz-ard functions [7] which proposed the Poisson-exponentialdistribution and [8] which proposed the complementaryexponential geometric distribution which is complementaryto the exponential geometric distribution proposed by [1]The last two proposed distributions accommodate increasinghazard functions

In this paper following [8] we propose a new distributionfamily by extending the exponentiated exponential distribu-tion [2] by compounding it with a geometric distributionhereafter the complementary exponentiated exponential geo-metric distribution or simplistically the CE2G distributionThe new distribution genesis is stated on a complementaryrisk problem base [9] in presence of latent risks in thesense that there is no information about which factor wasresponsible for the component failure and only themaximumlifetime value among all risks is observed This family haveone shape and two scale parameters accommodating increas-ing decreasing and bathtub failure rates

2 Journal of Probability and Statistics

The paper is organized as follows In Section 2 we intro-duce the new CE2G distribution derive the expressions forthe probability density survival and hazard functions and the119901th quantile and present its genesis In Section 3 we presentsome of its properties such as its characteristic function119903th raw moment mean and variance order statistics 119903thmoment of the 119894th order statistic mean residual lifetimeand modal value In Section 8 we present the inferentialprocedure In Section 10 the practical importance of the newdistribution was demonstrated in three applications whereour distribution outperforms several former lifetime distri-butions such as the exponential the exponential-geometricthe Weibull the modified Weibull and the generalizedexponential Poisson distribution Some final comments inSection 11 conclude the paper

2 The CE2G Model

Let119884 be a nonnegative random variable denoting the lifetimeof a component in some population The random variable 119884is said to have a CE2G distribution with parameters 120582 gt 0120572 gt 0 and 0 lt 120579 lt 1 if its probability density function (pdf)is given by

119891 (119910) =120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[1 minus (1 minus 120579) (1 minus 119890minus120582119910)120572

]2

119910 gt 0 (1)

where 120582 is a scale parameter of the distribution and 120572 and 120579are shape parameters Figure 1(a) shows the CE2Gprobabilitydensity function for 120582 = 1 120579 = 005 05 095 and 120572 =

03 10 3 and we can see that the function can be decreasingor unimodal

The survival function of a CE2G distributed randomvariable is given by

119878 (119910) =1 minus (1 minus 119890

minus120582119910

)120572

1 minus (1 minus 120579) (1 minus 119890minus120582119910)120572

119910 gt 0 (2)

where 120572 gt 0 120579 isin (0 1) and 120582 gt 0From (2) and (1) the failure rate function according to

the relationship ℎ(119910) = 119891(119910)119878(119910) is given by

ℎ (119910) =120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[1 minus (1 minus 119890minus120582119910)120572

] [1 minus (1 minus 120579) (1 minus 119890minus120582119910)120572

] (3)

The initial value is not finite if 120572 lt 1 and otherwise isgiven by ℎ(0) = 120582120579 if 120572 = 1 or ℎ(0) = 0 if 120572 gt 1 and thelong-term hazard function value is ℎ(infin) = 120582 The failurerate (3) can be increasing decreasing or bathtub as shownin Figure 1(b) which shows some failure rate function shapesto 120582 = 1 120579 = 005 05 095 and 120572 = 03 10 3

The 119901th quantile of the CE2G distribution is given by

119876 (119906) = 119865minus1

(119906) = minusln (1 minus (119906 (120579 (1 minus 119906) + 119906))1120572)

120582 (4)

where 119906 has the uniform 119880(0 1) distribution and 119865(119910) = 1 minus119878(119910) is the distribution function of 119884

Consider that in the study of reliability we can observeonly the maximum component lifetime for each componentamong all risks On many occasions the information aboutwhat risk produces the dead of the component in analysis isnot available or it is impossible that the true cause of failureis specified Complementary risks (CR) problems arise inseveral areas and an extensive literature is available Interestedreaders can see [10ndash12]

Then in this context ourmodel can be derived as followsLet119872 be a random variable denoting the number of failurecauses 119898 = 1 2 and considering 119872 with geometricalprobability distribution given by

119875 (119872 = 119898) = 120579(1 minus 120579)119898minus1

(5)

where 0 lt 120579 lt 1 and119872 = 1 2 Also consider 119905

119894

119894 = 1 2 3 realizations of a randomvariable denoting the failure times that is the time-to-eventdue to the 119894th CR and from [2] 119879

119894

has an exponentiatedexponential probability distribution with parameters 120582 and120572 given by

119891 (119905119894

120582 120572) = 120572119892 (119905119894

120582) 119866 (119905119894

120582)

= 120572120582 exp minus120582119905119894

(1 minus exp minus120582119905119894

)120572minus1

(6)

where 119892(sdot) and 119866(sdot) are the pdf and df respectively of theexponential distribution with parameter 120582

In the latent complementary risks scenario the number ofcauses119872 and the lifetime 119905

119895

associatedwith a particular causeare not observable (latent variables) but only the maximumlifetime 119884 among all causes is usually observed So we onlyobserve the random variable given by

119884 = max 1198791

1198792

119879119872

(7)

The following result shows that the randomvariable119884 hasprobability density function given by (1)

Proposition 1 If the random variable 119884 is defined as (7)then considering (5) and (6) 119884 is distributed accordingto a CE2G distribution with probability density functiongiven by (1)

Proof The conditional density function of (7) given119872 = 119898

is given by

119891 (119910 | 119872 = 119898 120582 120572)

= 119898120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[(1 minus 119890minus120582119910

)120572

]119898minus1

119905 gt 0 119898 = 1

(8)

Journal of Probability and Statistics 3

0

01

02

03

04

05

Den

sity

Den

sity

0 2 4 6 8 10 12

0

02

04

06

08

1

0

02

04

06

08

1

Times

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

Den

sity

α = 03

λ = 1 θ = 005 λ = 1 θ = 05 λ = 1 θ = 095

α = 1α = 3

α = 03α = 1α = 3

α = 03α = 1α = 3

(a)

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

0

05

1

15

2

Haz

ard

func

tion

0

05

1

15

2

Haz

ard

func

tion

0

05

1

15

2

Haz

ard

func

tion

λ = 1 θ = 005 λ = 1 θ = 05 λ = 1 θ = 095

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 03α = 1α = 3

(b)

Figure 1 (a) Probability density function of the CE2G distribution (b) Failure rate function of the CE2G distribution We fixed 120582 = 1

Then the marginal probability density function of 119884 is givenby

119891 (119910) =

infin

sum

119898=1

119898120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[(1 minus 119890minus120582119910

)120572

]119898minus1

times 120579(1 minus 120579)119898minus1

= 120579120572120582119890minus120582119910

(1minus119890minus120582119910

)120572minus1

infin

sum

119898=1

119898[(1minus119890minus120582119910

)120572

(1minus120579)]119898minus1

= 120579120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

infin

sum

119898=1

[(1 minus 119890minus120582119910

)120572

(1 minus 120579)]119898minus1

1 minus (1 minus 119890minus120582119910)120572

(1 minus 120579)

= 120579120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[1


]

2

(9)

This completes the proof

3 Some Properties

Many of the most important features and characteristics ofa distribution can be studied through its moments such as

mean and variance A general expression for rth ordinarymoment 1205831015840

119903

= 119864(119884119903

) of the CE2G distribution is hard to beobtained and we resume the mean and variance as follows

The moment generating function of the 119884 variable withdensity function given by (1) can be obtained analytically ifwe consider the expression given in [13 page 329 Equation(16)]

int

1

0

119911119901minus1

(1 minus 119911)119899minus1

(1 + 119887119911119898

)119897

119889119911

= Γ (119899)

infin

sum

119896=0

(119897

119896)(119887)119896

Γ (119901 + 119896119898)

Γ (119901 + 119899 + 119896119898)

(10)

For any real number 119905 let Φ119884

(119905) be the characteristicfunction of 119884 that is Φ

119884

(119905) = 119864[119890119894119905119884

] where 119894 denotes theimaginary unit With the preceding notations we state thefollowing

Proposition 2 For the random variable 119884 with CE2G distri-bution we have that its characteristic function is given by

Φ (119905) = 120572120579Γ (1 minus119894119905

120582)

infin

sum

119896=0

(minus2

119896)Γ (120572 [119896 + 1]) (120579 minus 1)

119896

Γ (120572 [119896 + 1] + 1 minus 119894119905120582)

(11)

where 119894 = radicminus1


Proof Consider the following

Φ119884

(119905) = int

infin

0

119890119894119905119910

119891 (119910) 119889119910

= int

infin

0

119890119894119905119910

120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

= 120572120579int

1

0

119911120572minus1

(1 minus 119911)minus119894119905120582

(1 minus (1 minus 120579) 119911120572)2

119889119911

(12)

where the last equality follows from the change of variable119911 = 1 minus 119890

minus120582119910Comparing the last integral with (10) obtaining 119899 = 1 minus

119894119905120582 119887 = 120579 minus 1 119898 = 120572 = 119901 and 119897 = minus2 and making theappropriate substitutions completed the proof

Proposition 3 A random variable 119884 with density given by (1)has mean and variance given respectively by

119864 (119884) =120579

120582

infin

sum

119896=0

(minus2

119896)(120579minus1)

119896

(119896+1)[Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)]

Var (119884) = 120579

1205822

infin

sum

119896=0

[(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

minus (Ψ(0 1)2

+1205872

6+ Ψ (0 120572 [119896 + 1] + 1)

times [Ψ (0 120572 [119896+1]+1) minus 2Ψ (0 1)]

minusΨ (1 120572 [119896 + 1] + 1) )]

minus 120579[

infin

sum

119896=0

(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

times (Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)) ]

2

(13)

whereΨ(119899 119911) = (119889119899+1119889119911119899+1) ln(Γ(119911)) is known as PsiGammafunction

Proof The first result follows from the relationshipΦ1015840

119884

(119905)119894|119905=0

= 119864(119884) From the literature Φ10158401015840119884

(119905)1198942

|119905=0

= 119864(1198842

)

and Var(119884) = 119864(1198842)minus [119864(119884)]2 and with a little algebra followthe results

Skewness is ameasure of the asymmetry of the probabilitydistribution The skewness value can be positive or negativeor even undefined Qualitatively a negative skew indicatesthat the tail on the left side of the probability density functionis longer than the right side and the bulk of the values lie tothe right of the mean A positive skew indicates that the tailon the right side is longer than the left side and the bulk of the

values lie to the left of the mean The skewness of a randomvariable 119884 say 120574

1

is given by the third standardized moment

1205741

=119864 [(119884 minus 120583)

3

]

(119864 [(119884 minus 120583)2

])32

=119864 (1198843

) minus 3119864 (1198842

) 119864 (119884) + 31198642

(119884) 119864 (119884) minus 1198643

(119884)

[119864 (1198842) minus 1198642 (119884)]32

(14)Kurtosis is any measure of the ldquopeakednessrdquo of the

probability distribution of a real-valued random variableIn a similar way to the concept of skewness kurtosis is adescriptor of the shape of a probability distribution It iscommon practice to use the kurtosis to provide a comparisonof the shape of a given distribution to that of the normaldistribution One common measure of kurtosis originatingwith Karl Pearson say 120574

2

is based on a scaled version of thefourth moment given by

1205742

=119864 [(119884 minus 120583)

4

]

(119864 [(119884 minus 120583)2

])2

=119864 (1198844

) minus 4119864 (1198843

) 119864 (119884) + 6119864 (1198842

) 1198642

(119884) minus 31198644

(119884)

[119864 (1198842) minus 1198642 (119884)]2

(15)Algebraic expressions of kurtosis and skewness are exten-

sive to show due to the fact that is necessary the alge-braic moment expressions up order four This moment canbe obtained by algebraic manipulation to determine 119864(119884)119864(1198842

)119864(1198843) and119864(1198844) in (14) and (15) through the Equation(11) Figure 2 shows the kurtosis (120574

2

) and skewness (1205741

) of theCE2G distribution for 120572 with 120582 = 1 120579 = 01 05 09 and for 120579with 120582 = 1 120572 = 03 10 3

4 Order Statistics

Order statistics are among the most fundamental tools innonparametric statistics and inference Let 119884

1

119884119899

bea random sample taken from the CE2G distribution and1198841119899

119884119899119899

denote the corresponding order statistics Thenthe pdf 119891

119894119899

(119910) of the 119894th order statistics 119884119894119899

is given by

119891119894119899

(119909) =119899

(119896 minus 1) (119899 minus 119896)119865(119910)119896minus1

(1 minus 119865 (119910))119899minus119896

119891 (119910)

(16)The 119903th moment of the 119894th order statistic 119884

119894119899

can beobtained from the following result due to [14]

119864 [119884119903

119894119899

] =119903

119899

sum

119901=119899minus119894+1

(minus1)119901minus119899+119894minus1

(119901 minus 1

119899 minus 119894)(119899

119901)int

infin

0

119910119903minus1

[119878 (119910)]119901

119889119910

(17)Consider the binomial series expansion given by

(1 minus 119909)minus119903

=

infin

sum

119896=0

(119903)119896

119896119909119896

(18)


0 04 080

100200300400500

Kur

tosi

s

0 04 08minus15minus10minus5

05

1015

minus15minus10minus5

05

1015

Skew

ness

0 1 2 30

100200300400500

Kur

tosi

s

0 1 2 3

Skew

ness

λ = 1 λ = 1 λ = 1 λ = 1

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(a)

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0 04 08 0 04 08 0 1 2 3 0 1 2 3

λ = 2 λ = 2 λ = 2 λ = 2

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(b)

Figure 2 (a) Kurtosis and skewness of CE2G distribution for fixed 120582 = 1 (b) Kurtosis and skewness of CE2G distribution for fixed 120582 = 2

where (119903)119896

is a Pochhammer symbol given (119903)119896

= 119903(119903 +

1) sdot sdot sdot (119903 + 119896 minus 1) and if |119909| lt 1 the series converge and

(minus119903)119896

= (minus1)119896

(119903 minus 119896 + 1)119896

(19)

Proposition 4 For the random variable 119884 with CE2G distri-bution we have that 119903th moment of the 119894th order statistic isgiven by

119864 [119884119903

119894119899

] =119903

120582119903

119899

sum

119901=119899minus119894+1

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0

(minus1)119901minus119899+119894+119903+119898+119897minus2

(119901 minus 1

119899 minus 119894)(

119899

119901)

times(1 minus 120579)

119895

(119901)119895

(119901 minus 119897+1)119897

(120572 (119895+119897)+119896 minus 119898 + 1)119898

119895119897119898(119898 + 1)119903

(20)

Proof From (2) and (18) we have that

int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910

= int

infin

0

119910119903minus1

(1 minus (1 minus 119890

minus120582119910

)120572


)

119901

119889119910

=(minus1)119903minus1

120582119903int

1

0

ln119903minus1 (1 minus 119909)(1 minus 119909)

(1 minus 119909120572

1 minus (1 minus 120579) 119909120572)

119901

119889119909


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times int

1

0

119909120572(119895+119897)+119896ln119903minus1 (1 minus 119909) 119889119909

(21)

Using the change of variable ln(1minus119909) = minus119906 and the expansion(18) results in the kernel of the gamma distribution functionas

int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times(minus[120572(119895 + 119897) + 119896])

119898

119898

(119903 minus 1)

(119898 + 1)119903

(22)

Now considering (22) in (17) and the property (19) the resultfollows

5 Entropy

An entropy of a randomvariable119884 is ameasure of variation ofthe uncertainty A popular entropy measure is Renyi entropy[15]


If 119884 has the probability density function (1) then Renyientropy is defined by

120574 (120588) =1

1 minus 120588log(int119891120588 (119910) 119889119910) (23)

where 120588 gt 0 and 120588 = 1

Proposition 5 If the randomvariable119884 is defined as (7) thenthe Renyi entropy is given by

120574 (120588) =1

1 minus 120588

timeslog((120572120579)120588120582120588minus1infin

sum

119896=0

[(1minus120579)119896

(2120588)119896

Γ (120588 (120572minus1)+119896120572+1)

timesΓ (120588) (119896Γ (120572 (120588+119896)+1))minus1

])

(24)

Proof From (23) we can calculate

int119891120588

(119910) 119889119910

= int

infin

0

(120572120582120579)120588

119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)

[1 minus (1 minus 120579)(1 minus 119890minus120582119910)120572

]2120588

119889119910

= (120572120582120579)120588

int

infin

0

infin

sum

119896=0

[119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

times (1 minus 120579)119896

(2120588)119896

119896] 119889119910

= (120572120579)120588

int

infin

0

infin

sum

119896=0

[(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

(1 minus 120579)119896

times(2120588)119896

119896(120582119890minus120582119910

)120588minus1

]120582119890minus120582119910

119889119910

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896int

infin

0

119906120588(120572minus1)+119896120572

times (1 minus 119906)120588minus1

119889119906]

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896

timesΓ (120588 (120572 minus 1) + 119896120572 + 1) Γ (120588)

Γ (120572 (120588 + 119896) + 1)]

(25)

So using the (25) in 120574(120588) the result follows

6 Reliability

In the context of reliability the stress-strength modeldescribes the life of a component which has a randomstrength 119884 that is subjected to a random stress 119883 Thecomponent fails at the instant hat the stress applied toit exceeds the strength and the component will functionsatisfactorily whenever 119884 gt 119883 So 119877 = Pr(119883 lt 119884) isa measure of component reliability In the area of stress-strength models there has been a large amount of workas regards estimation of the reliability 119877 when 119884 and 119883

are independent random variables belonging to the sameunivariate family of distributions

Proposition6 If the randomvariable119884 is defined as (7) thenthe reliability 119877 = 119875(119883 119884) for119883 and 119884 iid is given by

1205792

infin

sum

119896=0

(1 minus 120579)119896

(3)119896

119896 (119896 + 2) (26)

Proof For119883 and 119884 iid CE2G rvrsquos where119883 is the stress and119884 is the strength the reliability 119877 = 119875(119883 lt 119884) is given by

119877 = int

infin

0

int

119910

0

120572120582120579119890minus120582119909

(1 minus 119890minus120582119909

)120572minus1


]2

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119909 119889119910

= int

infin

0

120579(1 minus 119890minus120582119910

)120572


]

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

=

infin

sum

119896=0

1205792

120572120582(3)119896

119896(1 minus 120579)

119896

times int

infin

0

(1 minus 119890minus120582119910

)120572(119896+2)minus1

119890minus120582119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572120582(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895(1 minus 120579)

119896

times int

infin

0

119890minus120582(119895+1)119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895 (119895 + 1)(1 minus 120579)

119896

=

infin

sum

119896=0

1205792

(3)119896

119896 (119896 + 2)(1 minus 120579)

119896

(27)



7 Residual Lifetime Distribution

Given that there was no failure prior to time 119905 the residuallifetime distribution of a random variable 119883 distributed asCE2G distribution has the survival function given by

119878119905

(119909) = Pr [119883 gt 119909 + 119905 | 119883 gt 119905]

= (1 minus (1 minus 119890

minus120582(119909+119905)

)120572


)

times (1 minus (1 minus 120579) (1 minus 119890

minus120582119905

)120572

1 minus (1 minus 120579) (1 minus 119890minus120582(119909+119905))120572

)

(28)

The mean residual lifetime of a continuous distributionwith survival function 119865(119909) is given by

120583 (119905) = 119864 (119883 minus 119905 | 119883 gt 119905) =1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 (29)

Proposition 7 For the random variable 119884 with CE2G distri-bution we have that the mean residual lifetime is given by

120583 (119905) =1

120582(1 minus (1 minus 120579) (1 minus 119890

minus120582119905

)120572


)

times

infin

sum

119896=0

infin

sum

119894=0

1

sum

119895=0

(1 minus 120579)119894

(minus1)119895

119895

times (1 minus (1 minus 119890

120582119905

)120572(119894+119895)+119896+1

120572 (119894 + 119895) + 119896 + 1)

(30)

Proof From (29) and using 119878(119910) given by (2) we have that

1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 =1 minus (1 minus 120579) (1 minus 119890

minus120582119905

)120572


times int

infin

119905

1 minus (1 minus 119890minus120582119906

)120572


119889119906

=1

120582

1 minus (1 minus 120579) (1 minus 119890minus120582119905

)


times int

1

1minus119890

minus120582119905

1 minus 119909120572

(1 minus 119909120572 (1 minus 120579)) (1 minus 119909)119889119909

(31)

Now using (18) andmaking a binomial expansion in a similarway of the proof of Proposition 4 on (22) the result follows

8 Inference

Assuming the lifetimes are independently distributed and areindependent from the censoring mechanism the maximumlikelihood estimates (MLEs) of the parameters are obtainedby direct maximization of the log-likelihood function givenby

ℓ (120579 120582 120572) = ln (120572120579120582)119899

sum

119894=1

119888119894

minus 120582

119899

sum

119894=1

119888119894

119910119894

+ (120572 minus 1)

119899

sum

119894=1

119888119894

ln (1 minus 119890minus120582119910119894)

+

119899

sum

119894=1

(1 minus 119888119894

) ln (1 minus (1 minus 119890minus120582119910119894)120572

)

minus

119899

sum

119894=1

(1 + 119888119894

) ln (1 minus (1 minus 120579) (1 minus 119890minus120582119910119894)120572

)

(32)

where 119888119894

is a censoring indicator which is equal to 0 or1 respectively if the data is censored or observed Theadvantage of this procedure is that it runs immediately usingexisting statistical packages We have considered the optimroutine of the R [16]

Large-sample inference for the parameters are based onthe MLEs and their estimated standard errors For (120572 120579 120582)we consider the observed Fisher informationmatrix given by

119868119865

(120572 120579 120582) = (

119868120572120572

119868120572120579

119868120572120582

119868120579120572

119868120579120579

119868120579120582

119868120582120572

119868120582120579

119868120582120582

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816(120572120579120582)=(

120579

120582)

(33)

where the elements of the matrix 119868119865

(120572 120579 120582) are given in theappendix

Under conditions that are fulfilled for the parameters120572 120579and 120582 in the interior of the parameter space the asymptoticdistribution of ( 120579 ) as 119899 rarr infin is a normal 3-variate withzero mean and variance covariance matrix 119868minus1

119865

(120572 120579 120582)In order to compare different distributions we relied

upon several authors in the literature for example [617ndash19] which use the Akaike information criterion (AIC)and Bayesian information criterion (BIC) values which aredefined respectively by minus2ℓ(sdot) + 2119902 and minus2ℓ(sdot) + 119902 log(119899)where ℓ(sdot) is the LogLikehood evaluated in the MLE vectoron respective distribution 119902 is the number of parametersestimated and 119899 is the sample size The best distributioncorresponds to a lower AIC and BIC values

9 Simulation Study

Regarding the performance of the MLEs in the process ofestimation a study was performed based on one hundredgenerated dataset from the CE2G with six different sets ofparameters for 119899 = 20 50 100 200 500 and 1000 In orderto have unbounded parameters we consider the followingrestrictions on the parameters in estimation process For


0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)

Figure 3 (a) Empirical TTT plot for the dataset 1198791 1198792 and 1198793 respectively (b) Models fitting for the dataset 1198791 1198792 and 1198793 respectively

the parameter 120579 we considered the transformation 120579 =

119890120579

lowast

(1 + 119890120579

lowast

) where 120579lowast isin R and for 120572 and 120582 consider anexponential transformation Based on the literature of theMLEs we can return on the original parameters thought ofthe transformations For the calculus of their variances weuse the delta method The values (120572 120582 120579) = (1 1 05) wereused as the initial values for all numerics simulations since120582 gt 0 120572 gt 0 and 0 lt 120579 lt 1

The results are condensated in Table 1 which shows theaverages of the MLEs Av( 120579) together with coverageprobability of the 95 confidence intervals for parameters of

the CE2G 119862(120572 120582 120579) the bias the mean squarer error MSEand their deviance Sd( 120579) These results suggest that theMLEs estimates have performed adequately The deviance oftheMLEs decrease when sample size increasesThe empiricalcoverage probabilities are close to the nominal coverage levelparticularly as sample size increases

10 Applications

In this section we compare the CE2G distribution fit withseveral usual lifetime distributions on three datasets extracted


Table1Meanof

theM

LEstheird

eviancescoveragesbiasa

ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)

(minus0001600129minus004

81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)


Table 2 Values of themdashmax ℓ(sdot) and AIC for all fitted distributions

E EE EG Weibull Gamma CE2G MW GEP BS BS-G1198791

AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390

from the literature The first dataset 1198791 refers to the serum-reversal time (days) of 143 children contaminated with HIVfrom vertical transmission at the university hospital of theRibeirao Preto Scholl of Medicine (Hospital das Clınicas daFaculdade de Medicina de Ribeirao Preto) from 1986 to 2001[20] Serum reversal can occur in children born frommothersinfected with HIV

The second dataset 1198792 is lifetimes in hours of 417 forty-watt 110-volt internally frosted incandescent lamps takenfrom 42 weekly quality control [21] Survival times in daysare given for the control group of lamps on original dataset

The third dataset 1198793 gives the survival times for labora-tory mice which were exposed to a fixed dose of radiationat an age of 5 to 6 weeks The cause of death for each mousewas determined after autopsy to be one of three possibilitiesthymic lymphoma (C1) reticulum cell sarcoma (C2) or othercauses (C3) [22] Consider here the minces of C3 in thecontrol group

Firstly in order to identify the shape of a lifetime datafailure rate function we will consider a graphical methodbased on the TTT plot [23] In its empirical version the TTTplot is given by 119866(119903119899) = [(sum119903

119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899

119894 = 1 119899 represent the orderstatistics of the sample It has been shown that the failure ratefunction is increasing (decreasing) if the TTT plot is concave(convex) Figure 3(a) shows concaveTTTplots for the11987911198792and 1198793 datasets indicating increasing failure rate functions

We compare the CE2G distribution fits with the expo-nential distribution with probability density function givenby 119891(119909) = 120582119890

minus120582119909 the exponentiated exponential distribu-tion EE with probability density function given by 119891(119909) =120572 lowast 120582119890

minus120582119909

(1 minus 119890minus120582119909

)120572minus1 the EG distribution [1] with prob-

ability density function given by 119891(119909) = 120582(1 minus (1 minus

120579)119890minus120582119909

)minus1 the Weibull distribution with probability density

function given by 119891(119909) = (120579120582)(119909120582)120579minus1

119890minus(119909120582)

120579

wherethe shape parameter is 120579 and scale parameter is 120582 thegamma distribution with probability density function givenby 119891(119909) = (1120582120579Γ(120579))119909120579minus1119890minus119909120582 with shape parameter 120579 andscale parameter 120582 the modified Weibull (MW) distributionwith probability density function given by 119891(119909) = 120572119909120579minus1(120579 +120582119909)119890120582119909

119890minus120572119909

120579 exp120582119909 where 120572 120579 ge 0 and 120582 gt 0 the generalizedexponential Poisson (GEP) distribution [6] with probability

density function given by 119891(119909) = (120572120573120582(1 minus 119890minus120582

)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1

119890minus120582minus120573119909+120582 exp(minus120573119909) the generalized Birnbaum-

Saunders (BS-G) distribution [24] with probability densityfunction given by119891(119910) = ((radic(119910 minus 120583)120573+radic120573(119909 minus 120583))2120572(119909minus120583))120601([radic(119910 minus 120583)120573 minus radic120573(119909 minus 120583)]120572) where 120601(sdot) is the prob-ability density distribution of the standard normal distri-bution and the Birnbaum-Saunders (BS) distribution TheBS distribution is obtained considering 120583 = 0 in the BS-Gprobability density function

Table 2 provides theAIC andBIC criterion values for eachdistribution They provide evidence in favor of our CE2Gdistribution for the datasets1198791 and1198792 in all of the three com-parison criterion For the dataset 1198793 the CE2G distributionprovides similar fitting to theWeibull andMWdistributionsimplying that the CE2G distribution is a competitor to theusual survival distributions These results are corroboratedby the empirical Kaplan-Meier survival functions and thefitted survival functions shown inFigure 3(b)TheMLEs (andtheir corresponding standard errors in parentheses) of theparameters 120572 120579(times1000) and 120582(times10000) of the CE2G dis-tribution are given respectively by 37469 (05688) 414860(97659) and 1753646 (71814) for 1198791 by 51765 (194159)02625 (09915) and 946676 (38720) for1198792 and by 00018180(09818) 00698 (03770) and 787704 (115084) for 1198793

11 Concluding Remarks

In this paper a new lifetime distribution is provided anddiscussed The CE2G distribution accommodates increasingdecreasing and bathtub failure rate functions and arises ina latent complementary risks scenario where the lifetimeassociated with a particular risk is not observable but onlythe maximum lifetime value among all risks The propertiesof the proposed distribution are discussed including a formalproof of its probability density function and explicit algebraicformulas for its survival and hazard functions moments 119903thmoment of the 119894th order statistic mean residual lifetimemodal value and the observed Fisher information matrixMaximum likelihood inference is implemented straightfor-wardly The practical importance of the new distributionwas demonstrated in three applications where the CE2Gdistribution provided the best fit in comparison with severalother former lifetime distributions


Appendix

In this appendix we show the values of the elements of theobserved Fisher information matrix in (33) From (32) weobtain

119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894

minus120572 (1minus119888

119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments

V Marchi and F Louzada are supported by the Brazilianorganizations CAPES and CNPq respectively The authorsare grateful to Dr Gauss Cordeiro Editor of this specialissue in as well as to the anonymous Referees for their com-ments criticisms and suggestions which lead to importantimprovements

References

[1] K Adamidis and S Loukas ldquoA lifetime distribution withdecreasing failure raterdquo Statistics amp Probability Letters vol 39no 1 pp 35ndash42 1998

[2] R D Gupta and D Kundu ldquoExponentiated exponential familyan alternative to gamma andWeibull distributionsrdquo BiometricalJournal vol 43 no 1 pp 117ndash130 2001

[3] R D Gupta and D Kundu ldquoGeneralized exponential distribu-tionsrdquo Australian amp New Zealand Journal of Statistics vol 41no 2 pp 173ndash188 1999

[4] S Nadarajah and S Kotz ldquoThe exponentiated type distribu-tionsrdquoActa ApplicandaeMathematicae vol 92 no 2 pp 97ndash1112006

[5] C Kus ldquoA new lifetime distributionrdquo Computational Statisticsamp Data Analysis vol 51 no 9 pp 4497ndash4509 2007

[6] W Barreto-Souza and F Cribari-Neto ldquoA generalization ofthe exponential-Poisson distributionrdquo Statistics amp ProbabilityLetters vol 79 no 24 pp 2493ndash2500 2009

[7] F Louzada-Neto V G Cancho and G D C Barriga ldquoThePoisson-exponential distribution a Bayesian approachrdquo Journalof Applied Statistics vol 38 no 6 pp 1239ndash1248 2011

[8] F Louzada M Roman and V G Cancho ldquoThe complementaryexponential geometric distribution model properties and acomparison with its counterpartrdquo Computational Statistics ampData Analysis vol 55 no 8 pp 2516ndash2524 2011

[9] F Louzada-Neto ldquoPoly-hazard regression models for lifetimedatardquo Biometrics vol 55 no 4 pp 1121ndash1125 1999

[10] J F Lawless Statistical Models and Methods for Lifetime DataWiley Series in Probability and Statistics John Wiley amp SonsNew York NY USA 2nd edition 2003

[11] M J Crowder A C Kimber R L Smith and T J SweetingStatistical Analysis of Reliability Data Chapman amp Hall Lon-don UK 1991

[12] D R Cox and D OakesAnalysis of Survival Data Monographson Statistics andApplied Probability ChapmanampHall LondonUK 1984

[13] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Elsevier New York NY USA 7th edition 2007

[14] H M Barakat and Y H Abdelkader ldquoComputing the momentsof order statistics from nonidentical random variablesrdquo Statisti-cal Methods amp Applications vol 13 no 1 pp 15ndash26 2004

[15] A Renyi ldquoOn measures of entropy and informationrdquo in Pro-ceedings of the Fourth Berkeley Symposium on MathematicalStatistics and Probability vol 1 of Contributions to theTheory ofStatistics pp 547ndash561 University of California Press BerkeleyCalif USA 1961

[16] R Development Core Team R Foundation for Statistical Com-puting Vienna Austria 2010 httpwwwR-projectorg

[17] W Barreto-Souza A L de Morais and G M Cordeiro ldquoTheweibull-geometric distributionrdquo Journal of Statistical Computa-tion and Simulation vol 81 no 5 pp 645ndash657 2011


[18] K Adamidis T Dimitrakopoulou and S Loukas ldquoOn anextension of the exponential-geometric distributionrdquo Statisticsamp Probability Letters vol 73 no 3 pp 259ndash269 2005

[19] G S Mudholkar and D K Srivasta ldquoExponentiated weibullfamily a reanalysis of the bus-motor-failure datardquo Technomet-rics vol 37 no 4 pp 436ndash445 1995

[20] G S C Perdona and F Louzada-Neto ldquoA general hazard modelfor lifetime data in the presence of cure raterdquo Journal of AppliedStatistics vol 38 no 7 pp 1395ndash1405 2011

[21] D Davis ldquoAn analysis of some failure datardquo Journal of theAmerican Statistical Association vol 47 no 258 pp 113ndash1501952

[22] D G Hoel ldquoA representation of mortality data by competingrisksrdquo Biometrics vol 28 no 2 pp 475ndash488 1972

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

[24] Z W Birnbaum and S C Saunders ldquoA new family of lifedistributionsrdquo Journal of Applied Probability vol 6 pp 319ndash3271969

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The paper is organized as follows In Section 2 we intro-duce the new CE2G distribution derive the expressions forthe probability density survival and hazard functions and the119901th quantile and present its genesis In Section 3 we presentsome of its properties such as its characteristic function119903th raw moment mean and variance order statistics 119903thmoment of the 119894th order statistic mean residual lifetimeand modal value In Section 8 we present the inferentialprocedure In Section 10 the practical importance of the newdistribution was demonstrated in three applications whereour distribution outperforms several former lifetime distri-butions such as the exponential the exponential-geometricthe Weibull the modified Weibull and the generalizedexponential Poisson distribution Some final comments inSection 11 conclude the paper

2 The CE2G Model

Let119884 be a nonnegative random variable denoting the lifetimeof a component in some population The random variable 119884is said to have a CE2G distribution with parameters 120582 gt 0120572 gt 0 and 0 lt 120579 lt 1 if its probability density function (pdf)is given by

119891 (119910) =120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119910 gt 0 (1)

where 120582 is a scale parameter of the distribution and 120572 and 120579are shape parameters Figure 1(a) shows the CE2Gprobabilitydensity function for 120582 = 1 120579 = 005 05 095 and 120572 =

03 10 3 and we can see that the function can be decreasingor unimodal

The survival function of a CE2G distributed randomvariable is given by

119878 (119910) =1 minus (1 minus 119890

minus120582119910

)120572


119910 gt 0 (2)

where 120572 gt 0 120579 isin (0 1) and 120582 gt 0From (2) and (1) the failure rate function according to

the relationship ℎ(119910) = 119891(119910)119878(119910) is given by

ℎ (119910) =120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[1 minus (1 minus 119890minus120582119910)120572

] [1 minus (1 minus 120579) (1 minus 119890minus120582119910)120572

] (3)

The initial value is not finite if 120572 lt 1 and otherwise isgiven by ℎ(0) = 120582120579 if 120572 = 1 or ℎ(0) = 0 if 120572 gt 1 and thelong-term hazard function value is ℎ(infin) = 120582 The failurerate (3) can be increasing decreasing or bathtub as shownin Figure 1(b) which shows some failure rate function shapesto 120582 = 1 120579 = 005 05 095 and 120572 = 03 10 3

The 119901th quantile of the CE2G distribution is given by

119876 (119906) = 119865minus1

(119906) = minusln (1 minus (119906 (120579 (1 minus 119906) + 119906))1120572)

120582 (4)

where 119906 has the uniform 119880(0 1) distribution and 119865(119910) = 1 minus119878(119910) is the distribution function of 119884

Consider that in the study of reliability we can observeonly the maximum component lifetime for each componentamong all risks On many occasions the information aboutwhat risk produces the dead of the component in analysis isnot available or it is impossible that the true cause of failureis specified Complementary risks (CR) problems arise inseveral areas and an extensive literature is available Interestedreaders can see [10ndash12]

Then in this context ourmodel can be derived as followsLet119872 be a random variable denoting the number of failurecauses 119898 = 1 2 and considering 119872 with geometricalprobability distribution given by

119875 (119872 = 119898) = 120579(1 minus 120579)119898minus1

(5)

where 0 lt 120579 lt 1 and119872 = 1 2 Also consider 119905

119894

119894 = 1 2 3 realizations of a randomvariable denoting the failure times that is the time-to-eventdue to the 119894th CR and from [2] 119879

119894

has an exponentiatedexponential probability distribution with parameters 120582 and120572 given by

119891 (119905119894

120582 120572) = 120572119892 (119905119894

120582) 119866 (119905119894

120582)

= 120572120582 exp minus120582119905119894

(1 minus exp minus120582119905119894

)120572minus1

(6)

where 119892(sdot) and 119866(sdot) are the pdf and df respectively of theexponential distribution with parameter 120582

In the latent complementary risks scenario the number ofcauses119872 and the lifetime 119905

119895

associatedwith a particular causeare not observable (latent variables) but only the maximumlifetime 119884 among all causes is usually observed So we onlyobserve the random variable given by

119884 = max 1198791

1198792

119879119872

(7)

The following result shows that the randomvariable119884 hasprobability density function given by (1)

Proposition 1 If the random variable 119884 is defined as (7)then considering (5) and (6) 119884 is distributed accordingto a CE2G distribution with probability density functiongiven by (1)

Proof The conditional density function of (7) given119872 = 119898

is given by

119891 (119910 | 119872 = 119898 120582 120572)

= 119898120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[(1 minus 119890minus120582119910

)120572

]119898minus1

119905 gt 0 119898 = 1

(8)


0

01

02

03

04

05

Den

sity

Den

sity

0 2 4 6 8 10 12

0

02

04

06

08

1

0

02

04

06

08

1

Times

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

Den

sity

α = 03

λ = 1 θ = 005 λ = 1 θ = 05 λ = 1 θ = 095

α = 1α = 3

α = 03α = 1α = 3

α = 03α = 1α = 3

(a)

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

0

05

1

15

2

Haz

ard

func

tion

0

05

1

15

2

Haz

ard

func

tion

0

05

1

15

2

Haz

ard

func

tion

λ = 1 θ = 005 λ = 1 θ = 05 λ = 1 θ = 095

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 03α = 1α = 3

(b)



119891 (119910) =

infin

sum

119898=1

119898120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[(1 minus 119890minus120582119910

)120572

]119898minus1


= 120579120572120582119890minus120582119910

(1minus119890minus120582119910

)120572minus1

infin

sum

119898=1

119898[(1minus119890minus120582119910

)120572

(1minus120579)]119898minus1

= 120579120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

infin

sum

119898=1

[(1 minus 119890minus120582119910

)120572

(1 minus 120579)]119898minus1


(1 minus 120579)

= 120579120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[1


]

2

(9)


3 Some Properties



119903

= 119864(119884119903



int

1

0

119911119901minus1

(1 minus 119911)119899minus1

(1 + 119887119911119898

)119897

119889119911

= Γ (119899)

infin

sum

119896=0

(119897

119896)(119887)119896

Γ (119901 + 119896119898)

Γ (119901 + 119899 + 119896119898)

(10)



119884

(119905) = 119864[119890119894119905119884



Φ (119905) = 120572120579Γ (1 minus119894119905

120582)

infin

sum

119896=0

(minus2

119896)Γ (120572 [119896 + 1]) (120579 minus 1)

119896

Γ (120572 [119896 + 1] + 1 minus 119894119905120582)

(11)




Φ119884

(119905) = int

infin

0

119890119894119905119910

119891 (119910) 119889119910

= int

infin

0

119890119894119905119910

120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

= 120572120579int

1

0

119911120572minus1

(1 minus 119911)minus119894119905120582

(1 minus (1 minus 120579) 119911120572)2

119889119911

(12)





119864 (119884) =120579

120582

infin

sum

119896=0

(minus2

119896)(120579minus1)

119896

(119896+1)[Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)]

Var (119884) = 120579

1205822

infin

sum

119896=0

[(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

minus (Ψ(0 1)2

+1205872

6+ Ψ (0 120572 [119896 + 1] + 1)

times [Ψ (0 120572 [119896+1]+1) minus 2Ψ (0 1)]

minusΨ (1 120572 [119896 + 1] + 1) )]

minus 120579[

infin

sum

119896=0

(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

times (Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)) ]

2

(13)



119884

(119905)119894|119905=0


(119905)1198942

|119905=0

= 119864(1198842

)




1


1205741

=119864 [(119884 minus 120583)

3

]

(119864 [(119884 minus 120583)2

])32

=119864 (1198843

) minus 3119864 (1198842

) 119864 (119884) + 31198642

(119884) 119864 (119884) minus 1198643

(119884)

[119864 (1198842) minus 1198642 (119884)]32



2


1205742

=119864 [(119884 minus 120583)

4

]

(119864 [(119884 minus 120583)2

])2

=119864 (1198844

) minus 4119864 (1198843

) 119864 (119884) + 6119864 (1198842

) 1198642

(119884) minus 31198644

(119884)

[119864 (1198842) minus 1198642 (119884)]2




2



4 Order Statistics


1

119884119899


119884119899119899


119894119899


is given by

119891119894119899

(119909) =119899


(1 minus 119865 (119910))119899minus119896

119891 (119910)


119894119899


119864 [119884119903

119894119899

] =119903

119899

sum

119901=119899minus119894+1


(119901 minus 1

119899 minus 119894)(119899

119901)int

infin

0

119910119903minus1

[119878 (119910)]119901

119889119910


(1 minus 119909)minus119903

=

infin

sum

119896=0

(119903)119896

119896119909119896

(18)


0 04 080

100200300400500

Kur

tosi

s


05

1015


05

1015

Skew

ness

0 1 2 30

100200300400500

Kur

tosi

s

0 1 2 3

Skew

ness

λ = 1 λ = 1 λ = 1 λ = 1

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(a)

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0 04 08 0 04 08 0 1 2 3 0 1 2 3

λ = 2 λ = 2 λ = 2 λ = 2

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(b)


where (119903)119896


= 119903(119903 +


(minus119903)119896

= (minus1)119896

(119903 minus 119896 + 1)119896

(19)


119864 [119884119903

119894119899

] =119903

120582119903

119899

sum

119901=119899minus119894+1

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0


(119901 minus 1

119899 minus 119894)(

119899

119901)


119895

(119901)119895

(119901 minus 119897+1)119897

(120572 (119895+119897)+119896 minus 119898 + 1)119898

119895119897119898(119898 + 1)119903

(20)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910

= int

infin

0

119910119903minus1


minus120582119910

)120572


)

119901

119889119910


120582119903int

1

0


(1 minus 119909120572

1 minus (1 minus 120579) 119909120572)

119901

119889119909


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times int

1

0

119909120572(119895+119897)+119896ln119903minus1 (1 minus 119909) 119889119909

(21)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times(minus[120572(119895 + 119897) + 119896])

119898

119898

(119903 minus 1)

(119898 + 1)119903

(22)


5 Entropy




120574 (120588) =1

1 minus 120588log(int119891120588 (119910) 119889119910) (23)

where 120588 gt 0 and 120588 = 1


120574 (120588) =1

1 minus 120588


sum

119896=0

[(1minus120579)119896

(2120588)119896

Γ (120588 (120572minus1)+119896120572+1)

timesΓ (120588) (119896Γ (120572 (120588+119896)+1))minus1

])

(24)


int119891120588

(119910) 119889119910

= int

infin

0

(120572120582120579)120588

119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)


]2120588

119889119910

= (120572120582120579)120588

int

infin

0

infin

sum

119896=0

[119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

times (1 minus 120579)119896

(2120588)119896

119896] 119889119910

= (120572120579)120588

int

infin

0

infin

sum

119896=0

[(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

(1 minus 120579)119896

times(2120588)119896

119896(120582119890minus120582119910

)120588minus1

]120582119890minus120582119910

119889119910

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896int

infin

0

119906120588(120572minus1)+119896120572


119889119906]

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896

timesΓ (120588 (120572 minus 1) + 119896120572 + 1) Γ (120588)

Γ (120572 (120588 + 119896) + 1)]

(25)


6 Reliability




1205792

infin

sum

119896=0

(1 minus 120579)119896

(3)119896

119896 (119896 + 2) (26)


119877 = int

infin

0

int

119910

0

120572120582120579119890minus120582119909

(1 minus 119890minus120582119909

)120572minus1


]2

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119909 119889119910

= int

infin

0

120579(1 minus 119890minus120582119910

)120572


]

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

=

infin

sum

119896=0

1205792

120572120582(3)119896

119896(1 minus 120579)

119896

times int

infin

0

(1 minus 119890minus120582119910

)120572(119896+2)minus1

119890minus120582119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572120582(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895(1 minus 120579)

119896

times int

infin

0

119890minus120582(119895+1)119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895 (119895 + 1)(1 minus 120579)

119896

=

infin

sum

119896=0

1205792

(3)119896

119896 (119896 + 2)(1 minus 120579)

119896

(27)





119878119905

(119909) = Pr [119883 gt 119909 + 119905 | 119883 gt 119905]

= (1 minus (1 minus 119890

minus120582(119909+119905)

)120572


)


minus120582119905

)120572


)

(28)


120583 (119905) = 119864 (119883 minus 119905 | 119883 gt 119905) =1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 (29)


120583 (119905) =1


minus120582119905

)120572


)

times

infin

sum

119896=0

infin

sum

119894=0

1

sum

119895=0

(1 minus 120579)119894

(minus1)119895

119895


120582119905

)120572(119894+119895)+119896+1

120572 (119894 + 119895) + 119896 + 1)

(30)


1

119878 (119905)int

infin

119905


minus120582119905

)120572


times int

infin

119905


)120572


119889119906

=1

120582


)


times int

1

1minus119890

minus120582119905

1 minus 119909120572


(31)


8 Inference


ℓ (120579 120582 120572) = ln (120572120579120582)119899

sum

119894=1

119888119894

minus 120582

119899

sum

119894=1

119888119894

119910119894

+ (120572 minus 1)

119899

sum

119894=1

119888119894

ln (1 minus 119890minus120582119910119894)

+

119899

sum

119894=1

(1 minus 119888119894


)

minus

119899

sum

119894=1

(1 + 119888119894


)

(32)

where 119888119894



119868119865

(120572 120579 120582) = (

119868120572120572

119868120572120579

119868120572120582

119868120579120572

119868120579120579

119868120579120582

119868120582120572

119868120582120579

119868120582120582

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816(120572120579120582)=(

120579

120582)

(33)




119865



9 Simulation Study



0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)



119890120579

lowast

(1 + 119890120579

lowast




10 Applications



Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05

Den

sity

Den

sity

0 2 4 6 8 10 12

0

02

04

06

08

1

0

02

04

06

08

1

Times

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

Den

sity

α = 03

λ = 1 θ = 005 λ = 1 θ = 05 λ = 1 θ = 095

α = 1α = 3

α = 03α = 1α = 3

α = 03α = 1α = 3

(a)

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

0 2 4 6 8 10 12

Times

0

05

1

15

2

Haz

ard

func

tion

0

05

1

15

2

Haz

ard

func

tion

0

05

1

15

2

Haz

ard

func

tion

λ = 1 θ = 005 λ = 1 θ = 05 λ = 1 θ = 095

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 03α = 1α = 3

(b)



119891 (119910) =

infin

sum

119898=1

119898120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[(1 minus 119890minus120582119910

)120572

]119898minus1


= 120579120572120582119890minus120582119910

(1minus119890minus120582119910

)120572minus1

infin

sum

119898=1

119898[(1minus119890minus120582119910

)120572

(1minus120579)]119898minus1

= 120579120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

infin

sum

119898=1

[(1 minus 119890minus120582119910

)120572

(1 minus 120579)]119898minus1


(1 minus 120579)

= 120579120572120582119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1

[1


]

2

(9)


3 Some Properties



119903

= 119864(119884119903



int

1

0

119911119901minus1

(1 minus 119911)119899minus1

(1 + 119887119911119898

)119897

119889119911

= Γ (119899)

infin

sum

119896=0

(119897

119896)(119887)119896

Γ (119901 + 119896119898)

Γ (119901 + 119899 + 119896119898)

(10)



119884

(119905) = 119864[119890119894119905119884



Φ (119905) = 120572120579Γ (1 minus119894119905

120582)

infin

sum

119896=0

(minus2

119896)Γ (120572 [119896 + 1]) (120579 minus 1)

119896

Γ (120572 [119896 + 1] + 1 minus 119894119905120582)

(11)




Φ119884

(119905) = int

infin

0

119890119894119905119910

119891 (119910) 119889119910

= int

infin

0

119890119894119905119910

120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

= 120572120579int

1

0

119911120572minus1

(1 minus 119911)minus119894119905120582

(1 minus (1 minus 120579) 119911120572)2

119889119911

(12)





119864 (119884) =120579

120582

infin

sum

119896=0

(minus2

119896)(120579minus1)

119896

(119896+1)[Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)]

Var (119884) = 120579

1205822

infin

sum

119896=0

[(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

minus (Ψ(0 1)2

+1205872

6+ Ψ (0 120572 [119896 + 1] + 1)

times [Ψ (0 120572 [119896+1]+1) minus 2Ψ (0 1)]

minusΨ (1 120572 [119896 + 1] + 1) )]

minus 120579[

infin

sum

119896=0

(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

times (Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)) ]

2

(13)



119884

(119905)119894|119905=0


(119905)1198942

|119905=0

= 119864(1198842

)




1


1205741

=119864 [(119884 minus 120583)

3

]

(119864 [(119884 minus 120583)2

])32

=119864 (1198843

) minus 3119864 (1198842

) 119864 (119884) + 31198642

(119884) 119864 (119884) minus 1198643

(119884)

[119864 (1198842) minus 1198642 (119884)]32



2


1205742

=119864 [(119884 minus 120583)

4

]

(119864 [(119884 minus 120583)2

])2

=119864 (1198844

) minus 4119864 (1198843

) 119864 (119884) + 6119864 (1198842

) 1198642

(119884) minus 31198644

(119884)

[119864 (1198842) minus 1198642 (119884)]2




2



4 Order Statistics


1

119884119899


119884119899119899


119894119899


is given by

119891119894119899

(119909) =119899


(1 minus 119865 (119910))119899minus119896

119891 (119910)


119894119899


119864 [119884119903

119894119899

] =119903

119899

sum

119901=119899minus119894+1


(119901 minus 1

119899 minus 119894)(119899

119901)int

infin

0

119910119903minus1

[119878 (119910)]119901

119889119910


(1 minus 119909)minus119903

=

infin

sum

119896=0

(119903)119896

119896119909119896

(18)


0 04 080

100200300400500

Kur

tosi

s


05

1015


05

1015

Skew

ness

0 1 2 30

100200300400500

Kur

tosi

s

0 1 2 3

Skew

ness

λ = 1 λ = 1 λ = 1 λ = 1

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(a)

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0 04 08 0 04 08 0 1 2 3 0 1 2 3

λ = 2 λ = 2 λ = 2 λ = 2

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(b)


where (119903)119896


= 119903(119903 +


(minus119903)119896

= (minus1)119896

(119903 minus 119896 + 1)119896

(19)


119864 [119884119903

119894119899

] =119903

120582119903

119899

sum

119901=119899minus119894+1

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0


(119901 minus 1

119899 minus 119894)(

119899

119901)


119895

(119901)119895

(119901 minus 119897+1)119897

(120572 (119895+119897)+119896 minus 119898 + 1)119898

119895119897119898(119898 + 1)119903

(20)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910

= int

infin

0

119910119903minus1


minus120582119910

)120572


)

119901

119889119910


120582119903int

1

0


(1 minus 119909120572

1 minus (1 minus 120579) 119909120572)

119901

119889119909


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times int

1

0

119909120572(119895+119897)+119896ln119903minus1 (1 minus 119909) 119889119909

(21)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times(minus[120572(119895 + 119897) + 119896])

119898

119898

(119903 minus 1)

(119898 + 1)119903

(22)


5 Entropy




120574 (120588) =1

1 minus 120588log(int119891120588 (119910) 119889119910) (23)

where 120588 gt 0 and 120588 = 1


120574 (120588) =1

1 minus 120588


sum

119896=0

[(1minus120579)119896

(2120588)119896

Γ (120588 (120572minus1)+119896120572+1)

timesΓ (120588) (119896Γ (120572 (120588+119896)+1))minus1

])

(24)


int119891120588

(119910) 119889119910

= int

infin

0

(120572120582120579)120588

119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)


]2120588

119889119910

= (120572120582120579)120588

int

infin

0

infin

sum

119896=0

[119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

times (1 minus 120579)119896

(2120588)119896

119896] 119889119910

= (120572120579)120588

int

infin

0

infin

sum

119896=0

[(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

(1 minus 120579)119896

times(2120588)119896

119896(120582119890minus120582119910

)120588minus1

]120582119890minus120582119910

119889119910

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896int

infin

0

119906120588(120572minus1)+119896120572


119889119906]

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896

timesΓ (120588 (120572 minus 1) + 119896120572 + 1) Γ (120588)

Γ (120572 (120588 + 119896) + 1)]

(25)


6 Reliability




1205792

infin

sum

119896=0

(1 minus 120579)119896

(3)119896

119896 (119896 + 2) (26)


119877 = int

infin

0

int

119910

0

120572120582120579119890minus120582119909

(1 minus 119890minus120582119909

)120572minus1


]2

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119909 119889119910

= int

infin

0

120579(1 minus 119890minus120582119910

)120572


]

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

=

infin

sum

119896=0

1205792

120572120582(3)119896

119896(1 minus 120579)

119896

times int

infin

0

(1 minus 119890minus120582119910

)120572(119896+2)minus1

119890minus120582119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572120582(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895(1 minus 120579)

119896

times int

infin

0

119890minus120582(119895+1)119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895 (119895 + 1)(1 minus 120579)

119896

=

infin

sum

119896=0

1205792

(3)119896

119896 (119896 + 2)(1 minus 120579)

119896

(27)





119878119905

(119909) = Pr [119883 gt 119909 + 119905 | 119883 gt 119905]

= (1 minus (1 minus 119890

minus120582(119909+119905)

)120572


)


minus120582119905

)120572


)

(28)


120583 (119905) = 119864 (119883 minus 119905 | 119883 gt 119905) =1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 (29)


120583 (119905) =1


minus120582119905

)120572


)

times

infin

sum

119896=0

infin

sum

119894=0

1

sum

119895=0

(1 minus 120579)119894

(minus1)119895

119895


120582119905

)120572(119894+119895)+119896+1

120572 (119894 + 119895) + 119896 + 1)

(30)


1

119878 (119905)int

infin

119905


minus120582119905

)120572


times int

infin

119905


)120572


119889119906

=1

120582


)


times int

1

1minus119890

minus120582119905

1 minus 119909120572


(31)


8 Inference


ℓ (120579 120582 120572) = ln (120572120579120582)119899

sum

119894=1

119888119894

minus 120582

119899

sum

119894=1

119888119894

119910119894

+ (120572 minus 1)

119899

sum

119894=1

119888119894

ln (1 minus 119890minus120582119910119894)

+

119899

sum

119894=1

(1 minus 119888119894


)

minus

119899

sum

119894=1

(1 + 119888119894


)

(32)

where 119888119894



119868119865

(120572 120579 120582) = (

119868120572120572

119868120572120579

119868120572120582

119868120579120572

119868120579120579

119868120579120582

119868120582120572

119868120582120579

119868120582120582

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816(120572120579120582)=(

120579

120582)

(33)




119865



9 Simulation Study



0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)



119890120579

lowast

(1 + 119890120579

lowast




10 Applications



Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Φ119884

(119905) = int

infin

0

119890119894119905119910

119891 (119910) 119889119910

= int

infin

0

119890119894119905119910

120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

= 120572120579int

1

0

119911120572minus1

(1 minus 119911)minus119894119905120582

(1 minus (1 minus 120579) 119911120572)2

119889119911

(12)





119864 (119884) =120579

120582

infin

sum

119896=0

(minus2

119896)(120579minus1)

119896

(119896+1)[Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)]

Var (119884) = 120579

1205822

infin

sum

119896=0

[(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

minus (Ψ(0 1)2

+1205872

6+ Ψ (0 120572 [119896 + 1] + 1)

times [Ψ (0 120572 [119896+1]+1) minus 2Ψ (0 1)]

minusΨ (1 120572 [119896 + 1] + 1) )]

minus 120579[

infin

sum

119896=0

(minus2

119896)(120579 minus 1)

119896

(119896 + 1)

times (Ψ (0 120572 [119896+1]+1) minus Ψ (0 1)) ]

2

(13)



119884

(119905)119894|119905=0


(119905)1198942

|119905=0

= 119864(1198842

)




1


1205741

=119864 [(119884 minus 120583)

3

]

(119864 [(119884 minus 120583)2

])32

=119864 (1198843

) minus 3119864 (1198842

) 119864 (119884) + 31198642

(119884) 119864 (119884) minus 1198643

(119884)

[119864 (1198842) minus 1198642 (119884)]32



2


1205742

=119864 [(119884 minus 120583)

4

]

(119864 [(119884 minus 120583)2

])2

=119864 (1198844

) minus 4119864 (1198843

) 119864 (119884) + 6119864 (1198842

) 1198642

(119884) minus 31198644

(119884)

[119864 (1198842) minus 1198642 (119884)]2




2



4 Order Statistics


1

119884119899


119884119899119899


119894119899


is given by

119891119894119899

(119909) =119899


(1 minus 119865 (119910))119899minus119896

119891 (119910)


119894119899


119864 [119884119903

119894119899

] =119903

119899

sum

119901=119899minus119894+1


(119901 minus 1

119899 minus 119894)(119899

119901)int

infin

0

119910119903minus1

[119878 (119910)]119901

119889119910


(1 minus 119909)minus119903

=

infin

sum

119896=0

(119903)119896

119896119909119896

(18)


0 04 080

100200300400500

Kur

tosi

s


05

1015


05

1015

Skew

ness

0 1 2 30

100200300400500

Kur

tosi

s

0 1 2 3

Skew

ness

λ = 1 λ = 1 λ = 1 λ = 1

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(a)

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0 04 08 0 04 08 0 1 2 3 0 1 2 3

λ = 2 λ = 2 λ = 2 λ = 2

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(b)


where (119903)119896


= 119903(119903 +


(minus119903)119896

= (minus1)119896

(119903 minus 119896 + 1)119896

(19)


119864 [119884119903

119894119899

] =119903

120582119903

119899

sum

119901=119899minus119894+1

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0


(119901 minus 1

119899 minus 119894)(

119899

119901)


119895

(119901)119895

(119901 minus 119897+1)119897

(120572 (119895+119897)+119896 minus 119898 + 1)119898

119895119897119898(119898 + 1)119903

(20)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910

= int

infin

0

119910119903minus1


minus120582119910

)120572


)

119901

119889119910


120582119903int

1

0


(1 minus 119909120572

1 minus (1 minus 120579) 119909120572)

119901

119889119909


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times int

1

0

119909120572(119895+119897)+119896ln119903minus1 (1 minus 119909) 119889119909

(21)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times(minus[120572(119895 + 119897) + 119896])

119898

119898

(119903 minus 1)

(119898 + 1)119903

(22)


5 Entropy




120574 (120588) =1

1 minus 120588log(int119891120588 (119910) 119889119910) (23)

where 120588 gt 0 and 120588 = 1


120574 (120588) =1

1 minus 120588


sum

119896=0

[(1minus120579)119896

(2120588)119896

Γ (120588 (120572minus1)+119896120572+1)

timesΓ (120588) (119896Γ (120572 (120588+119896)+1))minus1

])

(24)


int119891120588

(119910) 119889119910

= int

infin

0

(120572120582120579)120588

119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)


]2120588

119889119910

= (120572120582120579)120588

int

infin

0

infin

sum

119896=0

[119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

times (1 minus 120579)119896

(2120588)119896

119896] 119889119910

= (120572120579)120588

int

infin

0

infin

sum

119896=0

[(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

(1 minus 120579)119896

times(2120588)119896

119896(120582119890minus120582119910

)120588minus1

]120582119890minus120582119910

119889119910

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896int

infin

0

119906120588(120572minus1)+119896120572


119889119906]

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896

timesΓ (120588 (120572 minus 1) + 119896120572 + 1) Γ (120588)

Γ (120572 (120588 + 119896) + 1)]

(25)


6 Reliability




1205792

infin

sum

119896=0

(1 minus 120579)119896

(3)119896

119896 (119896 + 2) (26)


119877 = int

infin

0

int

119910

0

120572120582120579119890minus120582119909

(1 minus 119890minus120582119909

)120572minus1


]2

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119909 119889119910

= int

infin

0

120579(1 minus 119890minus120582119910

)120572


]

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

=

infin

sum

119896=0

1205792

120572120582(3)119896

119896(1 minus 120579)

119896

times int

infin

0

(1 minus 119890minus120582119910

)120572(119896+2)minus1

119890minus120582119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572120582(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895(1 minus 120579)

119896

times int

infin

0

119890minus120582(119895+1)119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895 (119895 + 1)(1 minus 120579)

119896

=

infin

sum

119896=0

1205792

(3)119896

119896 (119896 + 2)(1 minus 120579)

119896

(27)





119878119905

(119909) = Pr [119883 gt 119909 + 119905 | 119883 gt 119905]

= (1 minus (1 minus 119890

minus120582(119909+119905)

)120572


)


minus120582119905

)120572


)

(28)


120583 (119905) = 119864 (119883 minus 119905 | 119883 gt 119905) =1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 (29)


120583 (119905) =1


minus120582119905

)120572


)

times

infin

sum

119896=0

infin

sum

119894=0

1

sum

119895=0

(1 minus 120579)119894

(minus1)119895

119895


120582119905

)120572(119894+119895)+119896+1

120572 (119894 + 119895) + 119896 + 1)

(30)


1

119878 (119905)int

infin

119905


minus120582119905

)120572


times int

infin

119905


)120572


119889119906

=1

120582


)


times int

1

1minus119890

minus120582119905

1 minus 119909120572


(31)


8 Inference


ℓ (120579 120582 120572) = ln (120572120579120582)119899

sum

119894=1

119888119894

minus 120582

119899

sum

119894=1

119888119894

119910119894

+ (120572 minus 1)

119899

sum

119894=1

119888119894

ln (1 minus 119890minus120582119910119894)

+

119899

sum

119894=1

(1 minus 119888119894


)

minus

119899

sum

119894=1

(1 + 119888119894


)

(32)

where 119888119894



119868119865

(120572 120579 120582) = (

119868120572120572

119868120572120579

119868120572120582

119868120579120572

119868120579120579

119868120579120582

119868120582120572

119868120582120579

119868120582120582

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816(120572120579120582)=(

120579

120582)

(33)




119865



9 Simulation Study



0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)



119890120579

lowast

(1 + 119890120579

lowast




10 Applications



Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 04 080

100200300400500

Kur

tosi

s


05

1015


05

1015

Skew

ness

0 1 2 30

100200300400500

Kur

tosi

s

0 1 2 3

Skew

ness

λ = 1 λ = 1 λ = 1 λ = 1

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(a)

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0100200300400500

Kur

tosi

s


05

1015

Skew

ness

0 04 08 0 04 08 0 1 2 3 0 1 2 3

λ = 2 λ = 2 λ = 2 λ = 2

α αθ θ

α = 03α = 1α = 3

α = 03α = 1α = 3

α = 01α = 05α = 09

α = 01α = 05α = 09

(b)


where (119903)119896


= 119903(119903 +


(minus119903)119896

= (minus1)119896

(119903 minus 119896 + 1)119896

(19)


119864 [119884119903

119894119899

] =119903

120582119903

119899

sum

119901=119899minus119894+1

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0


(119901 minus 1

119899 minus 119894)(

119899

119901)


119895

(119901)119895

(119901 minus 119897+1)119897

(120572 (119895+119897)+119896 minus 119898 + 1)119898

119895119897119898(119898 + 1)119903

(20)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910

= int

infin

0

119910119903minus1


minus120582119910

)120572


)

119901

119889119910


120582119903int

1

0


(1 minus 119909120572

1 minus (1 minus 120579) 119909120572)

119901

119889119909


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times int

1

0

119909120572(119895+119897)+119896ln119903minus1 (1 minus 119909) 119889119909

(21)


int

infin

0

119910119903minus1

[119878(119910)]119901

119889119910


120582119903

infin

sum

119895=0

infin

sum

119896=0

119901

sum

119897=0

infin

sum

119898=0

(1 minus 120579)119895

(119901)119895

(minus119901)119897

119895119897

times(minus[120572(119895 + 119897) + 119896])

119898

119898

(119903 minus 1)

(119898 + 1)119903

(22)


5 Entropy




120574 (120588) =1

1 minus 120588log(int119891120588 (119910) 119889119910) (23)

where 120588 gt 0 and 120588 = 1


120574 (120588) =1

1 minus 120588


sum

119896=0

[(1minus120579)119896

(2120588)119896

Γ (120588 (120572minus1)+119896120572+1)

timesΓ (120588) (119896Γ (120572 (120588+119896)+1))minus1

])

(24)


int119891120588

(119910) 119889119910

= int

infin

0

(120572120582120579)120588

119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)


]2120588

119889119910

= (120572120582120579)120588

int

infin

0

infin

sum

119896=0

[119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

times (1 minus 120579)119896

(2120588)119896

119896] 119889119910

= (120572120579)120588

int

infin

0

infin

sum

119896=0

[(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

(1 minus 120579)119896

times(2120588)119896

119896(120582119890minus120582119910

)120588minus1

]120582119890minus120582119910

119889119910

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896int

infin

0

119906120588(120572minus1)+119896120572


119889119906]

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896

timesΓ (120588 (120572 minus 1) + 119896120572 + 1) Γ (120588)

Γ (120572 (120588 + 119896) + 1)]

(25)


6 Reliability




1205792

infin

sum

119896=0

(1 minus 120579)119896

(3)119896

119896 (119896 + 2) (26)


119877 = int

infin

0

int

119910

0

120572120582120579119890minus120582119909

(1 minus 119890minus120582119909

)120572minus1


]2

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119909 119889119910

= int

infin

0

120579(1 minus 119890minus120582119910

)120572


]

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

=

infin

sum

119896=0

1205792

120572120582(3)119896

119896(1 minus 120579)

119896

times int

infin

0

(1 minus 119890minus120582119910

)120572(119896+2)minus1

119890minus120582119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572120582(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895(1 minus 120579)

119896

times int

infin

0

119890minus120582(119895+1)119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895 (119895 + 1)(1 minus 120579)

119896

=

infin

sum

119896=0

1205792

(3)119896

119896 (119896 + 2)(1 minus 120579)

119896

(27)





119878119905

(119909) = Pr [119883 gt 119909 + 119905 | 119883 gt 119905]

= (1 minus (1 minus 119890

minus120582(119909+119905)

)120572


)


minus120582119905

)120572


)

(28)


120583 (119905) = 119864 (119883 minus 119905 | 119883 gt 119905) =1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 (29)


120583 (119905) =1


minus120582119905

)120572


)

times

infin

sum

119896=0

infin

sum

119894=0

1

sum

119895=0

(1 minus 120579)119894

(minus1)119895

119895


120582119905

)120572(119894+119895)+119896+1

120572 (119894 + 119895) + 119896 + 1)

(30)


1

119878 (119905)int

infin

119905


minus120582119905

)120572


times int

infin

119905


)120572


119889119906

=1

120582


)


times int

1

1minus119890

minus120582119905

1 minus 119909120572


(31)


8 Inference


ℓ (120579 120582 120572) = ln (120572120579120582)119899

sum

119894=1

119888119894

minus 120582

119899

sum

119894=1

119888119894

119910119894

+ (120572 minus 1)

119899

sum

119894=1

119888119894

ln (1 minus 119890minus120582119910119894)

+

119899

sum

119894=1

(1 minus 119888119894


)

minus

119899

sum

119894=1

(1 + 119888119894


)

(32)

where 119888119894



119868119865

(120572 120579 120582) = (

119868120572120572

119868120572120579

119868120572120582

119868120579120572

119868120579120579

119868120579120582

119868120582120572

119868120582120579

119868120582120582

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816(120572120579120582)=(

120579

120582)

(33)




119865



9 Simulation Study



0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)



119890120579

lowast

(1 + 119890120579

lowast




10 Applications



Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574 (120588) =1

1 minus 120588log(int119891120588 (119910) 119889119910) (23)

where 120588 gt 0 and 120588 = 1


120574 (120588) =1

1 minus 120588


sum

119896=0

[(1minus120579)119896

(2120588)119896

Γ (120588 (120572minus1)+119896120572+1)

timesΓ (120588) (119896Γ (120572 (120588+119896)+1))minus1

])

(24)


int119891120588

(119910) 119889119910

= int

infin

0

(120572120582120579)120588

119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)


]2120588

119889119910

= (120572120582120579)120588

int

infin

0

infin

sum

119896=0

[119890minus120582120588119910

(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

times (1 minus 120579)119896

(2120588)119896

119896] 119889119910

= (120572120579)120588

int

infin

0

infin

sum

119896=0

[(1 minus 119890minus120582119910

)120588(120572minus1)+119896120572

(1 minus 120579)119896

times(2120588)119896

119896(120582119890minus120582119910

)120588minus1

]120582119890minus120582119910

119889119910

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896int

infin

0

119906120588(120572minus1)+119896120572


119889119906]

= (120572120579)120588

120582120588minus1

infin

sum

119896=0

[(1 minus 120579)119896

(2120588)119896

119896

timesΓ (120588 (120572 minus 1) + 119896120572 + 1) Γ (120588)

Γ (120572 (120588 + 119896) + 1)]

(25)


6 Reliability




1205792

infin

sum

119896=0

(1 minus 120579)119896

(3)119896

119896 (119896 + 2) (26)


119877 = int

infin

0

int

119910

0

120572120582120579119890minus120582119909

(1 minus 119890minus120582119909

)120572minus1


]2

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119909 119889119910

= int

infin

0

120579(1 minus 119890minus120582119910

)120572


]

times120572120582120579119890minus120582119910

(1 minus 119890minus120582119910

)120572minus1


]2

119889119910

=

infin

sum

119896=0

1205792

120572120582(3)119896

119896(1 minus 120579)

119896

times int

infin

0

(1 minus 119890minus120582119910

)120572(119896+2)minus1

119890minus120582119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572120582(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895(1 minus 120579)

119896

times int

infin

0

119890minus120582(119895+1)119910

119889119910

=

infin

sum

119896=0

infin

sum

119895=0

1205792

120572(3)119896

(1 minus 120572 (119896 + 2))119895

119896119895 (119895 + 1)(1 minus 120579)

119896

=

infin

sum

119896=0

1205792

(3)119896

119896 (119896 + 2)(1 minus 120579)

119896

(27)





119878119905

(119909) = Pr [119883 gt 119909 + 119905 | 119883 gt 119905]

= (1 minus (1 minus 119890

minus120582(119909+119905)

)120572


)


minus120582119905

)120572


)

(28)


120583 (119905) = 119864 (119883 minus 119905 | 119883 gt 119905) =1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 (29)


120583 (119905) =1


minus120582119905

)120572


)

times

infin

sum

119896=0

infin

sum

119894=0

1

sum

119895=0

(1 minus 120579)119894

(minus1)119895

119895


120582119905

)120572(119894+119895)+119896+1

120572 (119894 + 119895) + 119896 + 1)

(30)


1

119878 (119905)int

infin

119905


minus120582119905

)120572


times int

infin

119905


)120572


119889119906

=1

120582


)


times int

1

1minus119890

minus120582119905

1 minus 119909120572


(31)


8 Inference


ℓ (120579 120582 120572) = ln (120572120579120582)119899

sum

119894=1

119888119894

minus 120582

119899

sum

119894=1

119888119894

119910119894

+ (120572 minus 1)

119899

sum

119894=1

119888119894

ln (1 minus 119890minus120582119910119894)

+

119899

sum

119894=1

(1 minus 119888119894


)

minus

119899

sum

119894=1

(1 + 119888119894


)

(32)

where 119888119894



119868119865

(120572 120579 120582) = (

119868120572120572

119868120572120579

119868120572120582

119868120579120572

119868120579120579

119868120579120582

119868120582120572

119868120582120579

119868120582120582

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816(120572120579120582)=(

120579

120582)

(33)




119865



9 Simulation Study



0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)



119890120579

lowast

(1 + 119890120579

lowast




10 Applications



Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119878119905

(119909) = Pr [119883 gt 119909 + 119905 | 119883 gt 119905]

= (1 minus (1 minus 119890

minus120582(119909+119905)

)120572


)


minus120582119905

)120572


)

(28)


120583 (119905) = 119864 (119883 minus 119905 | 119883 gt 119905) =1

119878 (119905)int

infin

119905

119878 (119906) 119889119906 (29)


120583 (119905) =1


minus120582119905

)120572


)

times

infin

sum

119896=0

infin

sum

119894=0

1

sum

119895=0

(1 minus 120579)119894

(minus1)119895

119895


120582119905

)120572(119894+119895)+119896+1

120572 (119894 + 119895) + 119896 + 1)

(30)


1

119878 (119905)int

infin

119905


minus120582119905

)120572


times int

infin

119905


)120572


119889119906

=1

120582


)


times int

1

1minus119890

minus120582119905

1 minus 119909120572


(31)


8 Inference


ℓ (120579 120582 120572) = ln (120572120579120582)119899

sum

119894=1

119888119894

minus 120582

119899

sum

119894=1

119888119894

119910119894

+ (120572 minus 1)

119899

sum

119894=1

119888119894

ln (1 minus 119890minus120582119910119894)

+

119899

sum

119894=1

(1 minus 119888119894


)

minus

119899

sum

119894=1

(1 + 119888119894


)

(32)

where 119888119894



119868119865

(120572 120579 120582) = (

119868120572120572

119868120572120579

119868120572120582

119868120579120572

119868120579120579

119868120579120582

119868120582120572

119868120582120579

119868120582120582

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816(120572120579120582)=(

120579

120582)

(33)




119865



9 Simulation Study



0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)



119890120579

lowast

(1 + 119890120579

lowast




10 Applications



Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 1

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

TTT plot

0 02 04 06 08 1

TTT plot

0 02 04 06 08 1

TTT plot

G(nr)

nr

G(nr)

nr

G(nr)

nr

(a)

0

02

04

06

08

1

0

02

04

06

08

1

0

02

04

06

08

1

0 200 400 600 800 1000

Time

0 500 1000 1500

Time

0 200 400 600

Time

EEGCE2GGEPWeibull

GammaMWEEBSBS-G

S(t)

esti

mat

edS(t)

esti

mat

edS(t)

esti

mat

ed

(b)



119890120579

lowast

(1 + 119890120579

lowast




10 Applications



Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table1Meanof

theM

LEstheird


ndMSE

119899Av(120572120582120579)

Sd(120572120582120579)

Bias

MSE

119862(120572120582120579)

(120572120582120579)=(148310075)

20(15716344

9707522)

(078901120403327)

(009160349700022)

(0624713

65101096)

(099099080)

50(149023402607145)

(04478071030306

6)(0010203026minus

00355)

(019

870591100943)

(099099086)

100

(147653258907233)

(0268304964

02494)

(minus00035015

89minus

00267)

(007130269200623)

(099099091)

200

(1479831846

07379)

(0209003846

02176)

(minus000

0200846minus00121)

(004330153600470)

(099099097)

500

(147253161707361)

(015

840297701811)

(minus0007500617minus00139)

(002490091600326)

(099099099)

1000

(1502031116076

97)

(010

61018

32013

21)

(002200011600197)

(00116

0033400177)

(099099092)

(120572120582120579)=(125263024)

20(163892778304016)

(1030508411033

42)

(0388901483016

16)

(1202607224013

67)

(099099099)

50(148262700403459)

(073780597602589)

(0232600704010

59)

(059300358600776)

(099099099)

100

(13892265630304

6)(0554903699018

93)

(013

9200263006

46)

(032

42013

6200396)

(099099099)

200

(128692614302729)

(033

390252001229)

(00369minus0015700329)

(011170063100160)

(099099099)

500

(1260

92602902497)

(019

80014

4400632)

(00109minus0027100097)

(003890021400041)

(099099099)

1000

(126962624302479)

(016

210112300517)

(00196minus0005700079)

(002640012500027)

(099099099)

(120572120582120579)=(025063020)

20(038520655404163)

(0265802378033

76)

(013

520025402163)

(0088200566015

96)

(092099099)

50(02809064

000264

1)(012

64013

68019

73)

(00309001000064

1)(001680018600427)

(099099099)

100

(02935060

6402841)

(011620093101732)

(00435minus0023600841)

(00152000

9100368)

(099099099)

200

(026570635402246)

(0081000744

010

09)

(001570005400246)

(000670005500107)

(099099099)

500

(025690638802078)

(004290049200537)

(000690008800078)

(000190002500029)

(099099099)

1000

(02536063130204

4)(003070030300339)

(00036000130004

4)(0000

9000

0900012)

(099099099)

(120572120582120579)=(030060090)

20(032

580781708033)

(01165037

5002751)

(0025801817minus

00967)

(001410172300843)

(099099080)

50(0281306879076

39)

(006580201302639)

(minus0018700879minus01361)

(0004

60047900875)

(099099085)

100

(028690653508123)

(004890140602222)

(minus0013100535minus00877)

(000250022400566)

(099099093)

200

(029050632508364)

(0034300921015

53)

(minus000

9500325minus00636)

(00013000

9500279)

(099099097)

500

(030070611708884)

(00219006

47012

14)

(0000700117minus

00116)

(00005000

4300147)

(099099097)

1000

(029700605308821)

(0018400455010

03)

(minus0003000053minus00179)

(000030002100103)

(099099098)

(120572120582120579)=(050200040)

20(057482341304948)

(027900806603586)

(007480341300948)

(0082607606013

63)

(099099099)

50(0601920303053

48)

(02218044

6102941)

(010

190030301348)

(0059101979010

38)

(099099099)

100

(051002059204423)

(016

220317802465)

(001000059200423)

(002620103500620)

(099099099)

200

(053

072000

904503)

(010

910249101864

)(00307000

0900503)

(001270061400369)

(099099099)

500

(0504519

95404194)

(007270159401154)

(00045minus000

4600194)

(000530025200136)

(099099099)

1000

(050512007204034)

(004930100200598)

(000510007200034)

(000240010000036)

(099099098)

(120572120582120579)=(200025080)

20(215

990319906131)

(10176011120344

9)(015

9900699minus01869)

(1050800171015

27)

(099099079)

50(208260274307193)

(052

200052802874)

(0082600243minus

00807)

(027660003300883)

(099099088)

100

(199840262907519)

(044190041802711)


81)

(019

330001900751)

(099099087)

200

(203220256907808)

(0304

60027202050)

(00322000

69minus

00192)

(00929000

0800420)

(099099097)

500

(199450255207849)

(016

130021801783)

(minus0005500052minus00151)

(00258000

0500317)

(099099092)

1000

(196590252607774)

(013

580016001496)

(minus0034100026minus00226)

(00194000

0300227)

(099099096)




AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












AIC 17237 16572 17258 16305 16494 16160 16600 16593 19197 17085BIC 17267 16632 17317 16365 16553 16249 16689 16682 19256 17173

1198792

AIC 66498 57032 66518 55990 56059 55710 56647 57053 56483 56013BIC 66539 57113 66599 56071 56138 55831 56768 57174 56563 56134

1198793

AIC 5498 5382 5518 5303 5365 5306 5307 5403 5508 5340BIC 5515 5416 5552 5337 5398 5356 5357 5453 5541 5390





119894=1

119884119894119899

) + (119899 minus 119903)119884119903119899

](sum119899

119894=1

119884119894119899

)where 119903 = 1 119899 and 119884

119894119899




minus120582119909

(1 minus 119890minus120582119909



120579)119890minus120582119909



119890minus(119909120582)

120579


119890minus120572119909



)120572

)(1 minus

119890minus120582+120582 exp(minus120573119909)

)120572minus1







Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Appendix


119868120572120572

=

119899

sum

119894=1

(119888119894

1205722+(1 minus 119888119894

) 119871120572

119894

ln2 (119871119894

)

119877119894

+(1 minus 119888119894

) 1198712120572

119894

ln2 (119871119894

)

1198772119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

ln2 (119871119894

)

119879119894

minus(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

ln2 (119871119894

)

1198792119894

)

119868120572120579

= 119868120579120572

=

119899

sum

119894=1

((1 + 119888119894

) 119871120572

119894

ln (119871119894

)

119879119894

+(1 + 119888119894

) (1 minus 120579) 1198712120572

119894

ln (119871119894

)

1198792119894

)

119868120572120582

= 119868120582120572

=

119899

sum

119894=1

(minus119888119894

119883119894

119871119894

+120572 (1 minus 119888

119894

) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119877119894

+(1 minus 119888119894

) 119871120572

119894

119883119894

119871119894

119877119894

+120572 (1 minus 119888

119894

) 1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198772119894

minus120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

ln (119871119894

)119883119894

119871119894

119879119894

minus(1 + 119888119894

) (1 minus 120579) 119871120572

119894

119883119894

119871119894

119879119894

minus120572 (1 + 119888

119894

) (1 minus 120579)2

1198712120572

119894

ln (119871119894

)119883119894

119871119894

1198792119894

)

119868120579120579

=

119899

sum

119894=1

(119888119894

1205792minus(1 + 119888119894

) 1198712120572

119894

1198792119894

)

119868120579120582

= 119868120582120579

=

119899

sum

119894=1

(120572 (1 + 119888

119894

) 119871120572

119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 1198712120572

119894

119883119894

119871119894

1198792119894

)

119868120582120582

=

119899

sum

119894=1

(119888119894

1205822+(120572 minus 1) 119888

119894

119910119894

119883119894

119871119894

+(120572 minus 1) 119888

119894

1198832

119894

1198712119894


119894

) 119871120572

119894

119910119894

119883119894

119871119894

119877119894


119894

) 119871120572

119894

1198832

119894

(1minus120572)

1198712119894

119877119894

+1205722

(1 minus 119888119894

) 1198712120572

119894

1198832

119894

1198712119894

1198772119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

119910119894

119883119894

119871119894

119879119894

+120572 (1 + 119888

119894

) (1 minus 120579) 119871120572

119894

1198832

119894

(1 minus 120572)

1198712119894

119879119894

minus1205722

(1 + 119888119894

) (1 minus 120579)2

1198712120572

119894

1198832

119894

1198712119894

1198792119894

)

(A1)

where 119871119894

= 1 minus 119890minus120582119910

119894 119877119894

= 1 minus 119871120572

119894

119879119894

= 1 minus (1 minus 120579)119871120572

119894

and119883119894

= 119910119894

119890minus120582119910

119894

Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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