Hindawi Publishing CorporationISRN Probability and StatisticsVolume 2013 Article ID 965972 9 pageshttpdxdoiorg1011552013965972

Research ArticleMaximum Likelihood Estimator of AUC fora Bi-Exponentiated Weibull Model

Fazhe Chang1 and Lianfen Qian12

1 Department of Mathematical Sciences Florida Atlantic University Boca Raton FL 33431 USA2College of Mathematics and Information Science Wenzhou University Zhejiang 325035 China

Correspondence should be addressed to Lianfen Qian lqianfauedu

Received 29 April 2013 Accepted 25 July 2013

Academic Editors N Chernov V Makis M Montero and O Pons

Copyright copy 2013 F Chang and L QianThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For a Bi-Exponentiated Weibull model the authors obtain a general AUC formula and derive the maximum likelihood estimatorof AUC and its asymptotic property A simulation study is carried out to illustrate the finite sample size performance

1 ROC Curve and AUC for a Bi-ExponentiatedWeibull Model

In medical science a diagnostic test result called a biomarker[1 2] is an indicator for disease status of patients Theaccuracy of a medical diagnostic test is typically evaluated bysensitivity and specificity Receiver Operating Characteristic(ROC) curve is a graphical representation of the relationshipbetween sensitivity and specificity The area under the ROCcurve (AUC) is an overall performance measure for thebiomarker Hence the main issue in assessing the accuracy ofa diagnostic test is to estimate the ROC curve and its AUC

Suppose that there are two groups of study subjectsdiseased and nondiseased Let 119879 be a continuous biomarkerAssume that the larger 119879 is the more likely a subjectis diseased That is a subject is classified as positive ordiseased status if 119879 gt 119888 and as negative or nondiseased statusotherwise where 119888 is a cutoff point Let 119863 be the diseasestatus 119863 = 1 represents diseased population while 119863 = 0

represents nondiseased population Sensitivity of119879 is definedas the probability of being correctly classified as disease statusand specificity as the probability of being correctly classifiedas nondisease status That is

sensitivity (119888) = 119875 (119879 gt 119888 | 119863 = 1)

specificity (119888) = 119875 (119879 lt 119888 | 119863 = 0)

(1)

Let 119883 = 119879 | 119863 = 1 and 119884 = 119879 | 119863 = 0 be the biomarkersfor diseased and nondiseased subjects with survival functions1198781and 1198780 respectively Then

sensitivity (119888) = 1198781

(119888)

specificity (119888) = 1 minus 1198780

(119888)

(2)

The ROC function (curve) is defined as

ROC (119905) = 1198781

(119878minus1

0(119905)) where 119905 = 119878

0(119888) (3)

Notice that ROC curve is a monotonic increasing curve ina unit square starting at (0 0) and ending at (1 1) A goodbiomarker has a very concave down ROC curve

We say a bivariate random vector (119883 119884) is a Bi-model if119883 and 119884 are independent and are from the same distributionfamily but with different parameters Pepe [3] summarizesROC analysis and notices that among Bi-model Bi-normalmodel has been the most popular one The ROC curvemethod traces back to Green and Swets [4] in the signaldetection theory Bi-normal model assumes the existenceof a monotonic increasing transformation such that thebiomarker can be transformed into normally distributed forboth diseased and nondiseased patients Pepe [3 Results41 and 44] shows that the ROC curve of a biomarkeris invariant under a monotone increasing transformationHence a good estimator of ROC curve should satisfy thisinvariance property

2 ISRN Probability and Statistics

The literature for ROC analysis under Bi-normalmodel isextensive and the majority of the results are for a continuousbiomarker For a continuous biomarker Fushing and Turn-bull [5] obtained the estimators and their asymptotic proper-ties of the estimated ROC curve andAUCusing the empiricalbased nonparametric and semiparametric approaches for Bi-normalmodelMetz et al [6] studied properties ofmaximumlikelihood estimator via discretization over finite supportpoints Zou andHall [7] developed anMLE algorithm to esti-mate ROC curve under an unspecific or a specific monotonictransformation Gu et al [8] introduced a nonparametricmethod based on the Bayesian bootstrap technique to esti-mate ROC curve and compared the newmethod with severaldifferent semiparametric and nonparametric methods Pepeand Cai [9] considered the ROC curve as the distributionof placement value and then estimated the ROC curve bymaximizing the pseudolikelihood function of the estimatedplacement values All the abovemethods require complicatedcomputations Zhou andLin [10] proposed a semi-parametricmaximum likelihood (ML) estimator of ROC which satisfiesthe property of invariance Their method largely reduced thenumber of nuisance parameters and therefore the computa-tional complexity

In reality the distribution of the biomarker may beskewed and the normality assumption is not reasonableUnder this situation other models such as generalizedexponential (GE) Weibull (WE) and gamma to namea few provide a reasonable alternative see Faraggi et al[11] Another issue is that in the presence of two differentbiomarkers the one with uniformly higher ROC curve isbetter However when two ROC curves cross we cannotsimply compare and find which one is better Many differentnumerical indices have been proposed to summarize andcompare ROC curves of different biomarkers [12ndash15] Themost popular summary index is AUC defined as

AUC = int

1

0

ROC (119905) 119889119905 = 119875 (119883 gt 119884) (4)

The estimation of AUC = 119875(119883 gt 119884) is also of greatimportance in engineering reliability typically in stress-strength model For example if 119883 is the strength of a systemwhich is subject to a stress 119884 then 119875(119883 gt 119884) is a measure ofa system performance The system fails if and only if at anytime the applied stress is greater than its strength During thepast twenty years many results have been obtained on theestimation of the probability under the different parametricmodels Kotz et al [16] give a comprehensive summarizationabout stress-strength model

Many research results have been obtained for the estima-tion of AUC under popular survival models Bi-exponentialmodel [17] Bi-normal model [5 18] Bi-gamma model [19]Bi-Burr type X [20ndash23] Bi-generalized exponential [24] Bi-Weibull model [25] However all the popular survival modelssuch as generalized exponential (GE) Weibull (WE) andGamma do not allow nonmonotonic failure rates whichoften occur in real practice Furthermore the commonnonmonotonic failure rate in engineering and biologicalscience involves bathtub shapes

To overcome the aforementioned drawback many mod-els such as generalized gamma [26] generalized 119865 distribu-tion [27] two families [28] a four-parameter family [29]and a three-parameter (IDB) family [30] have been proposedfor modeling nonmonotone failure-rate data Mudholkarand Srivasta [31] proposed an exponentiated Weibull (EW)model an extension of both GE andWEmodels Mudholkarand Hutson [32] studied some properties for the model EWmodel not only includes distributions with unimodal andbathtub failure rates but also allows for a broader class ofmonotonic hazard rates EWmodelmay fit better thanGE forsome actual data For instance Khedhairi et al [33] concludethat generalized Rayleighmodel (a special case of EWmodel)is the best among the three models compared exponentialgeneralized exponential and generalized Rayleigh model

In the research of estimation of 119875(119883 gt 119884) under thedifferent survival models the most recent result in Baklizi[34] focuses the two-parameter Weibull model which is aspecial case of our Bi-EWmodel Baklizi assumes a commonshape parameter and different scale parameter which is alsostudied in Kundu and Gupta [25] In contrast to their studieshere we study the estimation of 119875(119883 gt 119884) under the Bi-EW model which shares one shape parameter only SinceEW includes Weibull as a submodel our results extend theones in Baklizi [34] and Kundu and Gupta [25] as shownin Theorem 1 On the other hand Baklizi [34] also studiedBayes estimator which we have not studied It will be a futureproject

In this paper we adopt AUC as a main summary index ofROCwhen comparing biomarkers under a Bi-ExponentiatedWeibull (Bi-EW) model We obtain a precise formula forthe AUC and derive the MLE of AUC and its asymptoticdistribution The remaining of this paper is organized asfollows Section 2 introduces the Bi-EW model and obtainsthe theoretical AUC formula Section 3 studies themaximumlikelihood estimation of AUC under the Bi-EW model andderives the asymptotic normality of the estimated AUCSection 4 reports the simulation results to compare theaccuracy of the estimated parameters and estimated AUC fordifferent sample sizes and various parameter settings in termsof absolute relative biases (ARB) and square root of meansquare error (RMSE) The conclusion is given in Section 5

2 A Bi-EW Model and Its AUC

A random variable 119883 has an Exponentiated Weibull (EW)distribution if its distribution function is defined as

119865 (119909 120572 120573 120582) = (1 minus 119890minus120582119909120573

)

120572

119909 gt 0 120572 gt 0 120573 gt 0 120582 gt 0

(5)

Denote 119883 sim EW(120572 120573 120582) Mudholkar and Srivasta [31]and Qian [35] show that this distribution allows bothmonotonic and bathtub shaped hazard rates The lattershape is useful in reality In this section we first derive ageneral moment formula for the distribution and then obtainAUC formula for a Bi-EW model with a common shapeparameter

ISRN Probability and Statistics 3

21 General Moment Formula of Exponentiated Weibull Tofully understand a new family of distributions it is importantto derive its summary properties such as moment formulas

Theorem 1 Let 119883 sim 119864119882(120572 120573 120582) One has

119864 [119883119894120573

(ln119883)119895]

= (minus1)119895 120572

120582119894120573119895

119895

sum

119897=0

(119895

119897) Γ(119897)

(119894 + 1)

times

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

[ln 120582 (119896 + 1)]119895minus119897

(119896 + 1)119894+1

(6)

where 119894 119895 = 0 1 2 3 and Γ(119894) = intinfin

0119890minus119905

119905119894minus1

119889119905

Proof By the definition of the EWmodel we have

119864 [119883119894120573

(ln119883)119895]

= 120572120582120573 int

infin

0

119909119894120573

(ln119909)119895(1 minus 119890

minus120582119909120573

)

120572minus1

times 119890minus120582119909120573

119909120573minus1

119889119909

= 120572120582120573

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

times int

infin

0

119909(119894+1)120573minus1

119890minus120582(119896+1)119909

120573

(ln119909)119895119889119909

(7)

By change of variables 119905 = 120582(119896 + 1)119909120573 we have

119889119909 =1

120573

119905(1120573)minus1

[120582 (119896 + 1)]1120573

119889119905 119909(119894+1)120573minus1

=119905119894+1minus1120573

[120582 (119896 + 1)]119894+1minus1120573

(8)

Then the integral part in (7) is equal to

int

infin

0

119890minus119905

(ln 119905 minus ln 120582 (119896 + 1)

120573)

1198951

120573

119905119894

[120582 (119896 + 1)]119894+1

119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

int

infin

0

119890minus119905

119905119894[ln 119905 minus ln 120582 (119896 + 1)]

119895119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897

times int

infin

0

119890minus119905

119905119894(ln 119905)119897119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897Γ(119897)

(119894 + 1)

(9)

Therefore we have

119864 [119883119894120573

(ln119883)119895]

= 120572120582120573

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

1

120573119895+1[120582 (119896 + 1)]119894+1

times

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897Γ(119897)

(119894 + 1)

= (minus1)119895 120572

120582119894120573119895

119895

sum

119897=0

(minus1)119897(

119895

119897) Γ(119897)

(119894 + 1)

times

infin

sum

119896=0

(120572 minus 1

119896)

[ln 120582 (119896 + 1)]119895minus119897

(119896 + 1)119894+1

(10)

Remark 2 Theorem 1 covers the following results as specialcases

Case 1 If 119895 = 0 and 119894120573 = 119896 then it reduces to the formulain Choudlhury [36] the simple moments of EW family Onehas

120583119896

= 119864 (119883119896) = 120572120582

minus119896120573Γ (

119896

120573+ 1)

times

120572minus1

sum

119894=0

(minus1)119894(

120572 minus 1

119894)

1

(119894 + 1)119896(120573+1)

(11)

Moreover if 119896120573minus1

= 119903 is positive integer then

120583119896

= 120572(minus120582)minus119903

[119889119903

119889119904119903119861(119904 120572)]

10038161003816100381610038161003816100381610038161003816119904=1

(12)

where 119861(119904 120572) = int1

0119909119904minus1

(1 minus 119909)120572minus1

119889119909

Case 2 If 119895 = 0 and 120573 = 1 it reducces to the moments ofGeneralized exponential family in Kunda and Gupta [24]

Case 3 If 119895 = 0 120572 = 1 and 119894120573 = 119899 it reduces to the momentsof Weibull family in Kunda and Gupta [25]

22 AUC under a Bi-EWModel In this subsection we derivethe general formula of AUC under a Bi-EW model with acommon shape parameter That is we assume that 119883 and119884 are independent and with the following EW distributionfunctions

1198651

(119909 1205721 120573 1205821) = (1 minus 119890

minus1205821119909120573

)

1205721

119909 gt 0 1205721

gt 0 120573 gt 0

1205821

gt 0

1198650

(119910 1205722 120573 1205822) = (1 minus 119890

minus1205822119910120573

)

1205722

119910 gt 0 1205722

gt 0 120573 gt 0

1205822

gt 0

(13)

4 ISRN Probability and Statistics

where the two distribution functions share a common shapeparameter 120573

Theorem3 Suppose that (119883 119884) follows the Bi-EWmodel withthe above distribution functions Then

119860119880119862 =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894 (14)

Proof By Taylor expansion we have

(1 minus 119890minus1205822119909120573

)

1205722

=

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

(1 minus 119890minus1205821119909120573

)

1205721

=

infin

sum

119895=0

(1205721

119895) (minus119890

minus1205821119909120573

)

119895

(15)

Thus

119860119880119862 = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821)

= int

infin

0

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

times

infin

sum

119895=1

(1205721

119895) 119895(minus119890

minus1205821119909120573

)

119895minus1

119890minus1205821119909120573

1205821120573119909120573minus1

119889119909

=

infin

sum

119894=0

infin

sum

119895=1

int

infin

0

(1205722

119894) (

1205721

119895) 119895(minus1)

119894+119895minus11205821119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus11205821119895 int

infin

0

119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894

(16)

Remark 4 (i) Note that the exact expression of AUC isindependent of the common parameter 120573 This is similar tothe cases in Bi-WE and Bi-GE models see Kundu and Gupta[24 25]

(ii) When 1205721

= 1205722

= 1 (14) reduces to the AUC for aBi-GE model

(iii) When 1205821

= 1205822 (14) reduces to the AUC for a Bi-WE

model To see (iii) we actually prove the following corollary

Corollary 5 For 1205721

gt 0 and 1205722

gt 0 one hasinfin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119895 + 119894=

1205721

1205721

+ 1205722

(17)

Proof Denote

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119905119894+119895minus1

119904119895119889119905 (18)

Then on one hand we have

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

(1205722

119894) (minus1)

119894119905119894minus1

infin

sum

119895=1

(1205721

119895) (minus1)

119895minus1(119905119904)119895119889119905

=120597

120597119904int

1

0

1

119905(1 minus 119905)1205722 (1 minus (1 minus 119905119904)

1205721)119889119905

= int

1

0

1

119905(1 minus 119905)

1205722(1 minus 119905119904)1205721minus11205721119905 119889119905

= 1205721

int

1

0

(1 minus 119905)1205722(1 minus 119905119904)

1205721minus1119889119905

(19)

On the other hand we have

119892 (119904) =120597

120597119904

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119904119895 119905119894+119895

119894 + 119895

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895119904119895minus1

119905119894+119895

(20)

Let 119904 = 1 then (19) implies that

119892 (1) =1205721

1205721

+ 1205722

(21)

while (20) implies that

119892 (1) =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895 (22)

This completes the proof of the corollary

3 Maximum Likelihood Estimator ofAUC and Its Asymptotic Property underthe Bi-EW Model

To estimate AUC under the Bi-EWmodel we adopt the plugin method That is we first obtain the maximum likelihoodestimator of the model parameters and then plug it into (14)to get the MLE of AUC

31 Maximum Likelihood Estimator of AUC under the Bi-EWModel Let119883

1 1198832 119883

119898and1198841 1198842 119884

119899be independent

random samples from 119883 sim 1198651and 119884 sim 119865

0 respectively

ISRN Probability and Statistics 5

Denote the model parameter vector by 120579 = (1205791 120579

5) =

(1205721 1205722 120573 1205821 1205822) Then the log-likelihood function is

119897 (120579) = 119898 ln1205721

+ 119898 ln 1205821

+ 119898 ln120573 + 119899 ln1205722

+ 119899 ln 1205822

+ 119899 ln120573 + (1205721

minus 1)

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 )

+ (1205722

minus 1)

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119894 )

minus

119898

sum

119894=1

1205821119909120573

119894+ (120573 minus 1)

119898

sum

119894=1

ln119909119894

minus

119899

sum

119895=1

1205822119910120573

119895+ (120573 minus 1)

119899

sum

119895=1

ln119910119894

(23)

We take derivatives with respect to the parameters to get thefollowing score equations

120597119897

1205971205721

=119898

1205721

+

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 ) = 0

120597119897

1205971205722

=119899

1205722

+

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119895 ) = 0

120597119897

1205971205821

=119898

1205821

minus

119898

sum

119894=1

119909120573

119894+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 119909120573

119894

1 minus 119890minus1205821119909120573119894

= 0

120597119897

1205971205822

=119899

1205822

minus

119899

sum

119895=1

119910120573

119895+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 119910120573

119895

1 minus 119890minus1205822119910120573119895

= 0

120597119897

120597120573=

119898 + 119899

120573minus 1205821

119898

sum

119894=1

119909120573

119894ln119909119894minus 1205822

119899

sum

119895=1

119910120573

119895ln119910119895

+

119898

sum

119894=1

ln119909119894+

119899

sum

119895=1

ln119910119895

+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 1205821119909120573

119894ln119909119894

1 minus 119890minus1205821119909120573119894

+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 1205822119910120573

119895ln119910119895

1 minus 119890minus1205822119910120573119895

= 0

(24)

The MLE 120579 of 120579 is the numerical solution of the above scoreequations Plugging 120579 into (14) we obtain the maximumlikelihood estimator of AUC as below

AUC =

infin

sum

119894=0

infin

sum

119895=1

(2

119894) (

1

119895) (minus1)

119894+119895minus1 1119895

1119895 + 2119894

(25)

32 Asymptotic Normality of the Estimated AUC In thissection we obtain the asymptotic distribution of 120579 = (

1 2

120573 1 2) and hence by the continuous mapping theorem

the asymptotic distribution of the estimated AUC We needto introduce some notation Let

120583119883

119895119896= 119864 (119883

120573(119895+1)(ln119883)

119896)

120583119884

119895119896= 119864 (119884

120573(119895+1)(ln119884)

119896) for 119895 119896 = 0 1 2

A (120579) = (

11988611

0 11988613

11988614

0

0 11988622

11988623

0 11988625

11988631

11988632

11988633

11988634

11988635

11988641

0 11988643

11988644

0

0 11988652

11988653

0 11988655

)

(26)

Then

11988611

=1

1205722

1

11988622

=1

1205722

2

11988613

= 11988631

= minus1205821

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))119895

119895120583119884

1198951

11988623

= 11988632

= minus1

radic1199011205822

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198951

11988625

= 11988652

= minus

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198950

11988633

= 1205821120583119883

02+

1

radic1199011205822120583119884

02+

1 + 119901

radic1199011205732

minus 1205821

(1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198952

minus1

radic1199011205822

(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))

119895(1 minus 120582

2(1 + 119894))

119895

120583119884

1198952

11988634

= 11988643

= 120583119883

01minus (1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198951

11988635

= 11988653

=1

radic119901120583119884

01minus

1

radic119901(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205822

(1 + 119894))

119895(minus1205822

(1 + 119894))119895

120583119884

1198951

11988644

=1

1205822

1

+ (1205721

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205821)119895

120583119883

119895+10

11988655

=1

1205822

2

+ (1205722

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205822)119895

120583119884

119895+10

(27)

6 ISRN Probability and Statistics

Estimated AUC

Freq

uenc

y

05 06 07 08

0

100

200

300

(a)

Estimated AUC

Freq

uenc

y

055 060 070 080

0

100

50

150

200

065 075

(b)

Estimated AUC

Freq

uenc

y

050 055 060 065 070 075 080

0

50

100

150

(c)

Estimated AUC

Freq

uenc

y

055 060 065 070 075

0

50

100

150

200

250

(d)

Figure 1 Histograms of the estimatedAUC for (a) (119898 119899) = (50 50) (b) (119898 119899) = (100 100) (c) (119898 119899) = (50 100) and (d) (119898 119899) = (100 200)

We denote 119863 = diag(radic119898 radic119899 radic119898 radic119898 radic119899) Noticethat EW family satisfies all the regularity conditions almosteverywhere Nowwe are ready to state the following theorem

Theorem 6 Suppose (119883 119884) follows the Bi-EWmodel with themodel parameter 120579 If 119898 rarr infin and 119899 rarr infin but 119898119899 rarr 119901for 0 lt 119901 lt infin then one has the following

(a) TheMLE 120579 of 120579 is asympotic normal

119863 (120579 minus 120579) 997888rarr 1198735

(0Aminus1 (120579)) (28)

(b) Part (a) the continuous mapping theorem and Deltamethod imply that

radic119898 (119860119880119862 minus 119860119880119862) 997888rarr 119873 (0 119861) (29)

where

119861 = (120597119860119880119862

120597120579)Aminus1 (120579) (

120597119860119880119862

120597120579)

119879

120597119860119880119862

120597120579= (

120597119860119880119862

1205971205721

120597119860119880119862

1205971205722

0120597119860119880119862

1205971205731

120597119860119880119862

1205971205732

)

(30)

4 A Simulation Study

In this section we present some results based on MonteCarlo simulation to check the performance of the MLE ofparameters and AUCWe consider small to moderate samplesizes 119898 119899 = 50 100 and 200 We set two scale parameters1205821

= 1205822

= 1 The common shape parameter is 120573 = 2The other shape parameters are 120572

1= 2 and 120572

2= 1 2 3 4

respectively The simulation is based on 1000 replicatesFigure 1 illustrates the histograms of the estimated AUC

under the sample sizes (119898 119899) = (50 50) (100 100) (50 100)

(100 200) when the true parameters are 1205721

= 2 1205722

= 4

ISRN Probability and Statistics 7

Table 1 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120573 = 2 120582

1= 1 and 120582

2= 1 and with

different values of 1205722

(119898 119899) MLE 1205722

= 1 1205722

= 2 1205722

= 3 1205722

= 4

(50 50)

1

03721 (16569) 035630 (2211) 03607 (15814) 03995 (1950)2

02556 (11068) 03645 (12601) 04204 (15814) 05547 (21276)120573 00801 (07525) 00675 (06655) 00615 (06679) 00525 (06243)1

00842 (05661) 00756 (05421) 0961 (05442) 00979 (05683)2

00913 (05235) 00805 (05493) 00916 (0560) 0119 (06189)AUC 00309 (00648) 00187 (00734) 00074 (008) 00078 (00648)

(50 100)

1

01872 (16205) 03035 (21506) 02333 (15885) 02205 (19536)2

00823 (04995) 02404 (18192) 02507 (26231) 02896 (21170)120573 00636 (05464) 00293 (05054) 00271 (04935) 00467 (05228)1

00348 (04380) 00985 (04777) 00859 (04421) 00653 (04614)2

00217 (03694) 00751 (04532) 00762 (04510) 00612 (04861)AUC 00072 (0060) 0005 (00591) 00037 (00565) 00058 (00424)

(100 50)

1

01609 (13638) 01689 (15535) 01774 (13754) 01731 (14751)2

01379 (05709) 02031 (16370) 02773 (19342) 03209 (17746)120573 00467 (05281) 00455 (04998) 00320 (04814) 00425 (05079)1

00399 (04085) 00430 (04129) 00602 (03962) 00461 (04191)2

00709 (04078) 00524 (04300) 00824 (04498) 00668 (04889)AUC 00048 (00519) 00031 (00489) 00017 (00469) 00056 (00435)

(100 100)

1

01205 (11180) 01322 (12528) 01396 (12052) 01276 (11109)2

00726 (04195) 01291 (12499) 01842 (21241) 01295 (20528)120573 00367 (04324) 00346 (04317) 00290 (04219) 00323 (04281)1

00318 (03521) 00354 (03687) 00469 (03720) 00369 (03644)2

0029 (03286) 00323 (03739) 00593 (03962) 00452 (03954)AUC 00036 (00412) 0003 (00458) 00018 (00331) 00031 (0040)

(200 200)

1

00668 (06871) 00569 (06666) 00700 (06368) 00602 (06553)2

00404 (02707) 00637 (06581) 00833 (11019) 00965 (07324)120573 00152 (02879) 00135 (02853) 00090 (02791) 00102 (02756)1

00214 (02487) 00215 (02457) 00324 (02399) 00217 (02430)2

00162 (02274) 00222 (02419) 00292 (02562) 00280 (02715)AUC 00021 (00256) 00033 (00267) 00031 (00270) 00013 (00264)

120573 = 2 and 1205821

= 1205822

= 1 Obviously as the sample sizeincreases the histogram tends to be more symmetric bellshaped Therefore the MLE of AUC performs well for themoderate sample size

We report the absolute relative biases (ARB) and thesquare root of mean squared errors (RMSEs) for the estima-tors of 120579 = (120579

1 120579

5) and AUC defined as below

ARB (120579119895) =

1

1000

1000

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

120579119894

119895minus 120579119895

120579119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

RMSE (120579119895) = radic

1

1000

1000

sum

119894=1

(120579119894

119895minus 120579119895)2

119895 = 1 2 3 4 5

(31)

where 120579119894

119895is the estimates of 120579

119895for the 119894th replicate

Recall that

AUC = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821) (32)

The estimator AUC and true value AUC can be computed byRiemann sums

119887

sum

119909=00001

1198650

(119909 2 120573 2) 1198911

(119909 1 120573 1) Δ119909

119887

sum

119909=00001

1198650

(119909 1205722 120573 1205822) 1198911

(119909 1205721 120573 1205821) Δ119909

(33)

respectively Here we choose 119887 = 4 and Δ119909 = 00001The interval [0 4] is divided evenly with each subintervallength 00001 and 119909 takes the value of the right end of eachsubintervalWe have verified that the summation keeps stablefor any larger value 119887 and smaller value Δ119909

Similarly

ARB (AUC) =1

1000

1000

sum

119894=1

1003816100381610038161003816100381610038161003816100381610038161003816

AUC119894minus AUC

AUC

1003816100381610038161003816100381610038161003816100381610038161003816

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRN Probability and Statistics

The literature for ROC analysis under Bi-normalmodel isextensive and the majority of the results are for a continuousbiomarker For a continuous biomarker Fushing and Turn-bull [5] obtained the estimators and their asymptotic proper-ties of the estimated ROC curve andAUCusing the empiricalbased nonparametric and semiparametric approaches for Bi-normalmodelMetz et al [6] studied properties ofmaximumlikelihood estimator via discretization over finite supportpoints Zou andHall [7] developed anMLE algorithm to esti-mate ROC curve under an unspecific or a specific monotonictransformation Gu et al [8] introduced a nonparametricmethod based on the Bayesian bootstrap technique to esti-mate ROC curve and compared the newmethod with severaldifferent semiparametric and nonparametric methods Pepeand Cai [9] considered the ROC curve as the distributionof placement value and then estimated the ROC curve bymaximizing the pseudolikelihood function of the estimatedplacement values All the abovemethods require complicatedcomputations Zhou andLin [10] proposed a semi-parametricmaximum likelihood (ML) estimator of ROC which satisfiesthe property of invariance Their method largely reduced thenumber of nuisance parameters and therefore the computa-tional complexity

In reality the distribution of the biomarker may beskewed and the normality assumption is not reasonableUnder this situation other models such as generalizedexponential (GE) Weibull (WE) and gamma to namea few provide a reasonable alternative see Faraggi et al[11] Another issue is that in the presence of two differentbiomarkers the one with uniformly higher ROC curve isbetter However when two ROC curves cross we cannotsimply compare and find which one is better Many differentnumerical indices have been proposed to summarize andcompare ROC curves of different biomarkers [12ndash15] Themost popular summary index is AUC defined as

AUC = int

1

0

ROC (119905) 119889119905 = 119875 (119883 gt 119884) (4)

The estimation of AUC = 119875(119883 gt 119884) is also of greatimportance in engineering reliability typically in stress-strength model For example if 119883 is the strength of a systemwhich is subject to a stress 119884 then 119875(119883 gt 119884) is a measure ofa system performance The system fails if and only if at anytime the applied stress is greater than its strength During thepast twenty years many results have been obtained on theestimation of the probability under the different parametricmodels Kotz et al [16] give a comprehensive summarizationabout stress-strength model

Many research results have been obtained for the estima-tion of AUC under popular survival models Bi-exponentialmodel [17] Bi-normal model [5 18] Bi-gamma model [19]Bi-Burr type X [20ndash23] Bi-generalized exponential [24] Bi-Weibull model [25] However all the popular survival modelssuch as generalized exponential (GE) Weibull (WE) andGamma do not allow nonmonotonic failure rates whichoften occur in real practice Furthermore the commonnonmonotonic failure rate in engineering and biologicalscience involves bathtub shapes

To overcome the aforementioned drawback many mod-els such as generalized gamma [26] generalized 119865 distribu-tion [27] two families [28] a four-parameter family [29]and a three-parameter (IDB) family [30] have been proposedfor modeling nonmonotone failure-rate data Mudholkarand Srivasta [31] proposed an exponentiated Weibull (EW)model an extension of both GE andWEmodels Mudholkarand Hutson [32] studied some properties for the model EWmodel not only includes distributions with unimodal andbathtub failure rates but also allows for a broader class ofmonotonic hazard rates EWmodelmay fit better thanGE forsome actual data For instance Khedhairi et al [33] concludethat generalized Rayleighmodel (a special case of EWmodel)is the best among the three models compared exponentialgeneralized exponential and generalized Rayleigh model

In the research of estimation of 119875(119883 gt 119884) under thedifferent survival models the most recent result in Baklizi[34] focuses the two-parameter Weibull model which is aspecial case of our Bi-EWmodel Baklizi assumes a commonshape parameter and different scale parameter which is alsostudied in Kundu and Gupta [25] In contrast to their studieshere we study the estimation of 119875(119883 gt 119884) under the Bi-EW model which shares one shape parameter only SinceEW includes Weibull as a submodel our results extend theones in Baklizi [34] and Kundu and Gupta [25] as shownin Theorem 1 On the other hand Baklizi [34] also studiedBayes estimator which we have not studied It will be a futureproject

In this paper we adopt AUC as a main summary index ofROCwhen comparing biomarkers under a Bi-ExponentiatedWeibull (Bi-EW) model We obtain a precise formula forthe AUC and derive the MLE of AUC and its asymptoticdistribution The remaining of this paper is organized asfollows Section 2 introduces the Bi-EW model and obtainsthe theoretical AUC formula Section 3 studies themaximumlikelihood estimation of AUC under the Bi-EW model andderives the asymptotic normality of the estimated AUCSection 4 reports the simulation results to compare theaccuracy of the estimated parameters and estimated AUC fordifferent sample sizes and various parameter settings in termsof absolute relative biases (ARB) and square root of meansquare error (RMSE) The conclusion is given in Section 5

2 A Bi-EW Model and Its AUC

A random variable 119883 has an Exponentiated Weibull (EW)distribution if its distribution function is defined as

119865 (119909 120572 120573 120582) = (1 minus 119890minus120582119909120573

)

120572

119909 gt 0 120572 gt 0 120573 gt 0 120582 gt 0

(5)

Denote 119883 sim EW(120572 120573 120582) Mudholkar and Srivasta [31]and Qian [35] show that this distribution allows bothmonotonic and bathtub shaped hazard rates The lattershape is useful in reality In this section we first derive ageneral moment formula for the distribution and then obtainAUC formula for a Bi-EW model with a common shapeparameter

ISRN Probability and Statistics 3

21 General Moment Formula of Exponentiated Weibull Tofully understand a new family of distributions it is importantto derive its summary properties such as moment formulas

Theorem 1 Let 119883 sim 119864119882(120572 120573 120582) One has

119864 [119883119894120573

(ln119883)119895]

= (minus1)119895 120572

120582119894120573119895

119895

sum

119897=0

(119895

119897) Γ(119897)

(119894 + 1)

times

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

[ln 120582 (119896 + 1)]119895minus119897

(119896 + 1)119894+1

(6)

where 119894 119895 = 0 1 2 3 and Γ(119894) = intinfin

0119890minus119905

119905119894minus1

119889119905

Proof By the definition of the EWmodel we have

119864 [119883119894120573

(ln119883)119895]

= 120572120582120573 int

infin

0

119909119894120573

(ln119909)119895(1 minus 119890

minus120582119909120573

)

120572minus1

times 119890minus120582119909120573

119909120573minus1

119889119909

= 120572120582120573

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

times int

infin

0

119909(119894+1)120573minus1

119890minus120582(119896+1)119909

120573

(ln119909)119895119889119909

(7)

By change of variables 119905 = 120582(119896 + 1)119909120573 we have

119889119909 =1

120573

119905(1120573)minus1

[120582 (119896 + 1)]1120573

119889119905 119909(119894+1)120573minus1

=119905119894+1minus1120573

[120582 (119896 + 1)]119894+1minus1120573

(8)

Then the integral part in (7) is equal to

int

infin

0

119890minus119905

(ln 119905 minus ln 120582 (119896 + 1)

120573)

1198951

120573

119905119894

[120582 (119896 + 1)]119894+1

119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

int

infin

0

119890minus119905

119905119894[ln 119905 minus ln 120582 (119896 + 1)]

119895119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897

times int

infin

0

119890minus119905

119905119894(ln 119905)119897119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897Γ(119897)

(119894 + 1)

(9)

Therefore we have

119864 [119883119894120573

(ln119883)119895]

= 120572120582120573

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

1

120573119895+1[120582 (119896 + 1)]119894+1

times

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897Γ(119897)

(119894 + 1)

= (minus1)119895 120572

120582119894120573119895

119895

sum

119897=0

(minus1)119897(

119895

119897) Γ(119897)

(119894 + 1)

times

infin

sum

119896=0

(120572 minus 1

119896)

[ln 120582 (119896 + 1)]119895minus119897

(119896 + 1)119894+1

(10)

Remark 2 Theorem 1 covers the following results as specialcases

Case 1 If 119895 = 0 and 119894120573 = 119896 then it reduces to the formulain Choudlhury [36] the simple moments of EW family Onehas

120583119896

= 119864 (119883119896) = 120572120582

minus119896120573Γ (

119896

120573+ 1)

times

120572minus1

sum

119894=0

(minus1)119894(

120572 minus 1

119894)

1

(119894 + 1)119896(120573+1)

(11)

Moreover if 119896120573minus1

= 119903 is positive integer then

120583119896

= 120572(minus120582)minus119903

[119889119903

119889119904119903119861(119904 120572)]

10038161003816100381610038161003816100381610038161003816119904=1

(12)

where 119861(119904 120572) = int1

0119909119904minus1

(1 minus 119909)120572minus1

119889119909

Case 2 If 119895 = 0 and 120573 = 1 it reducces to the moments ofGeneralized exponential family in Kunda and Gupta [24]

Case 3 If 119895 = 0 120572 = 1 and 119894120573 = 119899 it reduces to the momentsof Weibull family in Kunda and Gupta [25]

22 AUC under a Bi-EWModel In this subsection we derivethe general formula of AUC under a Bi-EW model with acommon shape parameter That is we assume that 119883 and119884 are independent and with the following EW distributionfunctions

1198651

(119909 1205721 120573 1205821) = (1 minus 119890

minus1205821119909120573

)

1205721

119909 gt 0 1205721

gt 0 120573 gt 0

1205821

gt 0

1198650

(119910 1205722 120573 1205822) = (1 minus 119890

minus1205822119910120573

)

1205722

119910 gt 0 1205722

gt 0 120573 gt 0

1205822

gt 0

(13)

4 ISRN Probability and Statistics

where the two distribution functions share a common shapeparameter 120573

Theorem3 Suppose that (119883 119884) follows the Bi-EWmodel withthe above distribution functions Then

119860119880119862 =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894 (14)

Proof By Taylor expansion we have

(1 minus 119890minus1205822119909120573

)

1205722

=

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

(1 minus 119890minus1205821119909120573

)

1205721

=

infin

sum

119895=0

(1205721

119895) (minus119890

minus1205821119909120573

)

119895

(15)

Thus

119860119880119862 = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821)

= int

infin

0

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

times

infin

sum

119895=1

(1205721

119895) 119895(minus119890

minus1205821119909120573

)

119895minus1

119890minus1205821119909120573

1205821120573119909120573minus1

119889119909

=

infin

sum

119894=0

infin

sum

119895=1

int

infin

0

(1205722

119894) (

1205721

119895) 119895(minus1)

119894+119895minus11205821119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus11205821119895 int

infin

0

119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894

(16)

Remark 4 (i) Note that the exact expression of AUC isindependent of the common parameter 120573 This is similar tothe cases in Bi-WE and Bi-GE models see Kundu and Gupta[24 25]

(ii) When 1205721

= 1205722

= 1 (14) reduces to the AUC for aBi-GE model

(iii) When 1205821

= 1205822 (14) reduces to the AUC for a Bi-WE

model To see (iii) we actually prove the following corollary

Corollary 5 For 1205721

gt 0 and 1205722

gt 0 one hasinfin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119895 + 119894=

1205721

1205721

+ 1205722

(17)

Proof Denote

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119905119894+119895minus1

119904119895119889119905 (18)

Then on one hand we have

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

(1205722

119894) (minus1)

119894119905119894minus1

infin

sum

119895=1

(1205721

119895) (minus1)

119895minus1(119905119904)119895119889119905

=120597

120597119904int

1

0

1

119905(1 minus 119905)1205722 (1 minus (1 minus 119905119904)

1205721)119889119905

= int

1

0

1

119905(1 minus 119905)

1205722(1 minus 119905119904)1205721minus11205721119905 119889119905

= 1205721

int

1

0

(1 minus 119905)1205722(1 minus 119905119904)

1205721minus1119889119905

(19)

On the other hand we have

119892 (119904) =120597

120597119904

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119904119895 119905119894+119895

119894 + 119895

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895119904119895minus1

119905119894+119895

(20)

Let 119904 = 1 then (19) implies that

119892 (1) =1205721

1205721

+ 1205722

(21)

while (20) implies that

119892 (1) =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895 (22)

This completes the proof of the corollary

3 Maximum Likelihood Estimator ofAUC and Its Asymptotic Property underthe Bi-EW Model

To estimate AUC under the Bi-EWmodel we adopt the plugin method That is we first obtain the maximum likelihoodestimator of the model parameters and then plug it into (14)to get the MLE of AUC

31 Maximum Likelihood Estimator of AUC under the Bi-EWModel Let119883

1 1198832 119883

119898and1198841 1198842 119884

119899be independent

random samples from 119883 sim 1198651and 119884 sim 119865

0 respectively

ISRN Probability and Statistics 5

Denote the model parameter vector by 120579 = (1205791 120579

5) =

(1205721 1205722 120573 1205821 1205822) Then the log-likelihood function is

119897 (120579) = 119898 ln1205721

+ 119898 ln 1205821

+ 119898 ln120573 + 119899 ln1205722

+ 119899 ln 1205822

+ 119899 ln120573 + (1205721

minus 1)

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 )

+ (1205722

minus 1)

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119894 )

minus

119898

sum

119894=1

1205821119909120573

119894+ (120573 minus 1)

119898

sum

119894=1

ln119909119894

minus

119899

sum

119895=1

1205822119910120573

119895+ (120573 minus 1)

119899

sum

119895=1

ln119910119894

(23)

We take derivatives with respect to the parameters to get thefollowing score equations

120597119897

1205971205721

=119898

1205721

+

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 ) = 0

120597119897

1205971205722

=119899

1205722

+

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119895 ) = 0

120597119897

1205971205821

=119898

1205821

minus

119898

sum

119894=1

119909120573

119894+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 119909120573

119894

1 minus 119890minus1205821119909120573119894

= 0

120597119897

1205971205822

=119899

1205822

minus

119899

sum

119895=1

119910120573

119895+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 119910120573

119895

1 minus 119890minus1205822119910120573119895

= 0

120597119897

120597120573=

119898 + 119899

120573minus 1205821

119898

sum

119894=1

119909120573

119894ln119909119894minus 1205822

119899

sum

119895=1

119910120573

119895ln119910119895

+

119898

sum

119894=1

ln119909119894+

119899

sum

119895=1

ln119910119895

+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 1205821119909120573

119894ln119909119894

1 minus 119890minus1205821119909120573119894

+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 1205822119910120573

119895ln119910119895

1 minus 119890minus1205822119910120573119895

= 0

(24)

The MLE 120579 of 120579 is the numerical solution of the above scoreequations Plugging 120579 into (14) we obtain the maximumlikelihood estimator of AUC as below

AUC =

infin

sum

119894=0

infin

sum

119895=1

(2

119894) (

1

119895) (minus1)

119894+119895minus1 1119895

1119895 + 2119894

(25)

32 Asymptotic Normality of the Estimated AUC In thissection we obtain the asymptotic distribution of 120579 = (

1 2

120573 1 2) and hence by the continuous mapping theorem

the asymptotic distribution of the estimated AUC We needto introduce some notation Let

120583119883

119895119896= 119864 (119883

120573(119895+1)(ln119883)

119896)

120583119884

119895119896= 119864 (119884

120573(119895+1)(ln119884)

119896) for 119895 119896 = 0 1 2

A (120579) = (

11988611

0 11988613

11988614

0

0 11988622

11988623

0 11988625

11988631

11988632

11988633

11988634

11988635

11988641

0 11988643

11988644

0

0 11988652

11988653

0 11988655

)

(26)

Then

11988611

=1

1205722

1

11988622

=1

1205722

2

11988613

= 11988631

= minus1205821

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))119895

119895120583119884

1198951

11988623

= 11988632

= minus1

radic1199011205822

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198951

11988625

= 11988652

= minus

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198950

11988633

= 1205821120583119883

02+

1

radic1199011205822120583119884

02+

1 + 119901

radic1199011205732

minus 1205821

(1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198952

minus1

radic1199011205822

(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))

119895(1 minus 120582

2(1 + 119894))

119895

120583119884

1198952

11988634

= 11988643

= 120583119883

01minus (1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198951

11988635

= 11988653

=1

radic119901120583119884

01minus

1

radic119901(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205822

(1 + 119894))

119895(minus1205822

(1 + 119894))119895

120583119884

1198951

11988644

=1

1205822

1

+ (1205721

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205821)119895

120583119883

119895+10

11988655

=1

1205822

2

+ (1205722

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205822)119895

120583119884

119895+10

(27)

6 ISRN Probability and Statistics

Estimated AUC

Freq

uenc

y

05 06 07 08

0

100

200

300

(a)

Estimated AUC

Freq

uenc

y

055 060 070 080

0

100

50

150

200

065 075

(b)

Estimated AUC

Freq

uenc

y

050 055 060 065 070 075 080

0

50

100

150

(c)

Estimated AUC

Freq

uenc

y

055 060 065 070 075

0

50

100

150

200

250

(d)

Figure 1 Histograms of the estimatedAUC for (a) (119898 119899) = (50 50) (b) (119898 119899) = (100 100) (c) (119898 119899) = (50 100) and (d) (119898 119899) = (100 200)

We denote 119863 = diag(radic119898 radic119899 radic119898 radic119898 radic119899) Noticethat EW family satisfies all the regularity conditions almosteverywhere Nowwe are ready to state the following theorem

Theorem 6 Suppose (119883 119884) follows the Bi-EWmodel with themodel parameter 120579 If 119898 rarr infin and 119899 rarr infin but 119898119899 rarr 119901for 0 lt 119901 lt infin then one has the following

(a) TheMLE 120579 of 120579 is asympotic normal

119863 (120579 minus 120579) 997888rarr 1198735

(0Aminus1 (120579)) (28)

(b) Part (a) the continuous mapping theorem and Deltamethod imply that

radic119898 (119860119880119862 minus 119860119880119862) 997888rarr 119873 (0 119861) (29)

where

119861 = (120597119860119880119862

120597120579)Aminus1 (120579) (

120597119860119880119862

120597120579)

119879

120597119860119880119862

120597120579= (

120597119860119880119862

1205971205721

120597119860119880119862

1205971205722

0120597119860119880119862

1205971205731

120597119860119880119862

1205971205732

)

(30)

4 A Simulation Study

In this section we present some results based on MonteCarlo simulation to check the performance of the MLE ofparameters and AUCWe consider small to moderate samplesizes 119898 119899 = 50 100 and 200 We set two scale parameters1205821

= 1205822

= 1 The common shape parameter is 120573 = 2The other shape parameters are 120572

1= 2 and 120572

2= 1 2 3 4

respectively The simulation is based on 1000 replicatesFigure 1 illustrates the histograms of the estimated AUC

under the sample sizes (119898 119899) = (50 50) (100 100) (50 100)

(100 200) when the true parameters are 1205721

= 2 1205722

= 4

ISRN Probability and Statistics 7

Table 1 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120573 = 2 120582

1= 1 and 120582

2= 1 and with

different values of 1205722

(119898 119899) MLE 1205722

= 1 1205722

= 2 1205722

= 3 1205722

= 4

(50 50)

1

03721 (16569) 035630 (2211) 03607 (15814) 03995 (1950)2

02556 (11068) 03645 (12601) 04204 (15814) 05547 (21276)120573 00801 (07525) 00675 (06655) 00615 (06679) 00525 (06243)1

00842 (05661) 00756 (05421) 0961 (05442) 00979 (05683)2

00913 (05235) 00805 (05493) 00916 (0560) 0119 (06189)AUC 00309 (00648) 00187 (00734) 00074 (008) 00078 (00648)

(50 100)

1

01872 (16205) 03035 (21506) 02333 (15885) 02205 (19536)2

00823 (04995) 02404 (18192) 02507 (26231) 02896 (21170)120573 00636 (05464) 00293 (05054) 00271 (04935) 00467 (05228)1

00348 (04380) 00985 (04777) 00859 (04421) 00653 (04614)2

00217 (03694) 00751 (04532) 00762 (04510) 00612 (04861)AUC 00072 (0060) 0005 (00591) 00037 (00565) 00058 (00424)

(100 50)

1

01609 (13638) 01689 (15535) 01774 (13754) 01731 (14751)2

01379 (05709) 02031 (16370) 02773 (19342) 03209 (17746)120573 00467 (05281) 00455 (04998) 00320 (04814) 00425 (05079)1

00399 (04085) 00430 (04129) 00602 (03962) 00461 (04191)2

00709 (04078) 00524 (04300) 00824 (04498) 00668 (04889)AUC 00048 (00519) 00031 (00489) 00017 (00469) 00056 (00435)

(100 100)

1

01205 (11180) 01322 (12528) 01396 (12052) 01276 (11109)2

00726 (04195) 01291 (12499) 01842 (21241) 01295 (20528)120573 00367 (04324) 00346 (04317) 00290 (04219) 00323 (04281)1

00318 (03521) 00354 (03687) 00469 (03720) 00369 (03644)2

0029 (03286) 00323 (03739) 00593 (03962) 00452 (03954)AUC 00036 (00412) 0003 (00458) 00018 (00331) 00031 (0040)

(200 200)

1

00668 (06871) 00569 (06666) 00700 (06368) 00602 (06553)2

00404 (02707) 00637 (06581) 00833 (11019) 00965 (07324)120573 00152 (02879) 00135 (02853) 00090 (02791) 00102 (02756)1

00214 (02487) 00215 (02457) 00324 (02399) 00217 (02430)2

00162 (02274) 00222 (02419) 00292 (02562) 00280 (02715)AUC 00021 (00256) 00033 (00267) 00031 (00270) 00013 (00264)

120573 = 2 and 1205821

= 1205822

= 1 Obviously as the sample sizeincreases the histogram tends to be more symmetric bellshaped Therefore the MLE of AUC performs well for themoderate sample size

We report the absolute relative biases (ARB) and thesquare root of mean squared errors (RMSEs) for the estima-tors of 120579 = (120579

1 120579

5) and AUC defined as below

ARB (120579119895) =

1

1000

1000

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

120579119894

119895minus 120579119895

120579119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

RMSE (120579119895) = radic

1

1000

1000

sum

119894=1

(120579119894

119895minus 120579119895)2

119895 = 1 2 3 4 5

(31)

where 120579119894

119895is the estimates of 120579

119895for the 119894th replicate

Recall that

AUC = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821) (32)

The estimator AUC and true value AUC can be computed byRiemann sums

119887

sum

119909=00001

1198650

(119909 2 120573 2) 1198911

(119909 1 120573 1) Δ119909

119887

sum

119909=00001

1198650

(119909 1205722 120573 1205822) 1198911

(119909 1205721 120573 1205821) Δ119909

(33)

respectively Here we choose 119887 = 4 and Δ119909 = 00001The interval [0 4] is divided evenly with each subintervallength 00001 and 119909 takes the value of the right end of eachsubintervalWe have verified that the summation keeps stablefor any larger value 119887 and smaller value Δ119909

Similarly

ARB (AUC) =1

1000

1000

sum

119894=1

1003816100381610038161003816100381610038161003816100381610038161003816

AUC119894minus AUC

AUC

1003816100381610038161003816100381610038161003816100381610038161003816

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 3

21 General Moment Formula of Exponentiated Weibull Tofully understand a new family of distributions it is importantto derive its summary properties such as moment formulas

Theorem 1 Let 119883 sim 119864119882(120572 120573 120582) One has

119864 [119883119894120573

(ln119883)119895]

= (minus1)119895 120572

120582119894120573119895

119895

sum

119897=0

(119895

119897) Γ(119897)

(119894 + 1)

times

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

[ln 120582 (119896 + 1)]119895minus119897

(119896 + 1)119894+1

(6)

where 119894 119895 = 0 1 2 3 and Γ(119894) = intinfin

0119890minus119905

119905119894minus1

119889119905

Proof By the definition of the EWmodel we have

119864 [119883119894120573

(ln119883)119895]

= 120572120582120573 int

infin

0

119909119894120573

(ln119909)119895(1 minus 119890

minus120582119909120573

)

120572minus1

times 119890minus120582119909120573

119909120573minus1

119889119909

= 120572120582120573

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

times int

infin

0

119909(119894+1)120573minus1

119890minus120582(119896+1)119909

120573

(ln119909)119895119889119909

(7)

By change of variables 119905 = 120582(119896 + 1)119909120573 we have

119889119909 =1

120573

119905(1120573)minus1

[120582 (119896 + 1)]1120573

119889119905 119909(119894+1)120573minus1

=119905119894+1minus1120573

[120582 (119896 + 1)]119894+1minus1120573

(8)

Then the integral part in (7) is equal to

int

infin

0

119890minus119905

(ln 119905 minus ln 120582 (119896 + 1)

120573)

1198951

120573

119905119894

[120582 (119896 + 1)]119894+1

119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

int

infin

0

119890minus119905

119905119894[ln 119905 minus ln 120582 (119896 + 1)]

119895119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897

times int

infin

0

119890minus119905

119905119894(ln 119905)119897119889119905

=1

120573119895+1[120582 (119896 + 1)]119894+1

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897Γ(119897)

(119894 + 1)

(9)

Therefore we have

119864 [119883119894120573

(ln119883)119895]

= 120572120582120573

infin

sum

119896=0

(minus1)119896

(120572 minus 1

119896)

1

120573119895+1[120582 (119896 + 1)]119894+1

times

119895

sum

119897=0

(119895

119897) [minus ln 120582 (119896 + 1)]

119895minus119897Γ(119897)

(119894 + 1)

= (minus1)119895 120572

120582119894120573119895

119895

sum

119897=0

(minus1)119897(

119895

119897) Γ(119897)

(119894 + 1)

times

infin

sum

119896=0

(120572 minus 1

119896)

[ln 120582 (119896 + 1)]119895minus119897

(119896 + 1)119894+1

(10)

Remark 2 Theorem 1 covers the following results as specialcases

Case 1 If 119895 = 0 and 119894120573 = 119896 then it reduces to the formulain Choudlhury [36] the simple moments of EW family Onehas

120583119896

= 119864 (119883119896) = 120572120582

minus119896120573Γ (

119896

120573+ 1)

times

120572minus1

sum

119894=0

(minus1)119894(

120572 minus 1

119894)

1

(119894 + 1)119896(120573+1)

(11)

Moreover if 119896120573minus1

= 119903 is positive integer then

120583119896

= 120572(minus120582)minus119903

[119889119903

119889119904119903119861(119904 120572)]

10038161003816100381610038161003816100381610038161003816119904=1

(12)

where 119861(119904 120572) = int1

0119909119904minus1

(1 minus 119909)120572minus1

119889119909

Case 2 If 119895 = 0 and 120573 = 1 it reducces to the moments ofGeneralized exponential family in Kunda and Gupta [24]

Case 3 If 119895 = 0 120572 = 1 and 119894120573 = 119899 it reduces to the momentsof Weibull family in Kunda and Gupta [25]

22 AUC under a Bi-EWModel In this subsection we derivethe general formula of AUC under a Bi-EW model with acommon shape parameter That is we assume that 119883 and119884 are independent and with the following EW distributionfunctions

1198651

(119909 1205721 120573 1205821) = (1 minus 119890

minus1205821119909120573

)

1205721

119909 gt 0 1205721

gt 0 120573 gt 0

1205821

gt 0

1198650

(119910 1205722 120573 1205822) = (1 minus 119890

minus1205822119910120573

)

1205722

119910 gt 0 1205722

gt 0 120573 gt 0

1205822

gt 0

(13)

4 ISRN Probability and Statistics

where the two distribution functions share a common shapeparameter 120573

Theorem3 Suppose that (119883 119884) follows the Bi-EWmodel withthe above distribution functions Then

119860119880119862 =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894 (14)

Proof By Taylor expansion we have

(1 minus 119890minus1205822119909120573

)

1205722

=

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

(1 minus 119890minus1205821119909120573

)

1205721

=

infin

sum

119895=0

(1205721

119895) (minus119890

minus1205821119909120573

)

119895

(15)

Thus

119860119880119862 = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821)

= int

infin

0

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

times

infin

sum

119895=1

(1205721

119895) 119895(minus119890

minus1205821119909120573

)

119895minus1

119890minus1205821119909120573

1205821120573119909120573minus1

119889119909

=

infin

sum

119894=0

infin

sum

119895=1

int

infin

0

(1205722

119894) (

1205721

119895) 119895(minus1)

119894+119895minus11205821119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus11205821119895 int

infin

0

119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894

(16)

Remark 4 (i) Note that the exact expression of AUC isindependent of the common parameter 120573 This is similar tothe cases in Bi-WE and Bi-GE models see Kundu and Gupta[24 25]

(ii) When 1205721

= 1205722

= 1 (14) reduces to the AUC for aBi-GE model

(iii) When 1205821

= 1205822 (14) reduces to the AUC for a Bi-WE

model To see (iii) we actually prove the following corollary

Corollary 5 For 1205721

gt 0 and 1205722

gt 0 one hasinfin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119895 + 119894=

1205721

1205721

+ 1205722

(17)

Proof Denote

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119905119894+119895minus1

119904119895119889119905 (18)

Then on one hand we have

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

(1205722

119894) (minus1)

119894119905119894minus1

infin

sum

119895=1

(1205721

119895) (minus1)

119895minus1(119905119904)119895119889119905

=120597

120597119904int

1

0

1

119905(1 minus 119905)1205722 (1 minus (1 minus 119905119904)

1205721)119889119905

= int

1

0

1

119905(1 minus 119905)

1205722(1 minus 119905119904)1205721minus11205721119905 119889119905

= 1205721

int

1

0

(1 minus 119905)1205722(1 minus 119905119904)

1205721minus1119889119905

(19)

On the other hand we have

119892 (119904) =120597

120597119904

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119904119895 119905119894+119895

119894 + 119895

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895119904119895minus1

119905119894+119895

(20)

Let 119904 = 1 then (19) implies that

119892 (1) =1205721

1205721

+ 1205722

(21)

while (20) implies that

119892 (1) =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895 (22)

This completes the proof of the corollary

3 Maximum Likelihood Estimator ofAUC and Its Asymptotic Property underthe Bi-EW Model

To estimate AUC under the Bi-EWmodel we adopt the plugin method That is we first obtain the maximum likelihoodestimator of the model parameters and then plug it into (14)to get the MLE of AUC

31 Maximum Likelihood Estimator of AUC under the Bi-EWModel Let119883

1 1198832 119883

119898and1198841 1198842 119884

119899be independent

random samples from 119883 sim 1198651and 119884 sim 119865

0 respectively

ISRN Probability and Statistics 5

Denote the model parameter vector by 120579 = (1205791 120579

5) =

(1205721 1205722 120573 1205821 1205822) Then the log-likelihood function is

119897 (120579) = 119898 ln1205721

+ 119898 ln 1205821

+ 119898 ln120573 + 119899 ln1205722

+ 119899 ln 1205822

+ 119899 ln120573 + (1205721

minus 1)

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 )

+ (1205722

minus 1)

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119894 )

minus

119898

sum

119894=1

1205821119909120573

119894+ (120573 minus 1)

119898

sum

119894=1

ln119909119894

minus

119899

sum

119895=1

1205822119910120573

119895+ (120573 minus 1)

119899

sum

119895=1

ln119910119894

(23)

We take derivatives with respect to the parameters to get thefollowing score equations

120597119897

1205971205721

=119898

1205721

+

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 ) = 0

120597119897

1205971205722

=119899

1205722

+

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119895 ) = 0

120597119897

1205971205821

=119898

1205821

minus

119898

sum

119894=1

119909120573

119894+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 119909120573

119894

1 minus 119890minus1205821119909120573119894

= 0

120597119897

1205971205822

=119899

1205822

minus

119899

sum

119895=1

119910120573

119895+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 119910120573

119895

1 minus 119890minus1205822119910120573119895

= 0

120597119897

120597120573=

119898 + 119899

120573minus 1205821

119898

sum

119894=1

119909120573

119894ln119909119894minus 1205822

119899

sum

119895=1

119910120573

119895ln119910119895

+

119898

sum

119894=1

ln119909119894+

119899

sum

119895=1

ln119910119895

+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 1205821119909120573

119894ln119909119894

1 minus 119890minus1205821119909120573119894

+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 1205822119910120573

119895ln119910119895

1 minus 119890minus1205822119910120573119895

= 0

(24)

The MLE 120579 of 120579 is the numerical solution of the above scoreequations Plugging 120579 into (14) we obtain the maximumlikelihood estimator of AUC as below

AUC =

infin

sum

119894=0

infin

sum

119895=1

(2

119894) (

1

119895) (minus1)

119894+119895minus1 1119895

1119895 + 2119894

(25)

32 Asymptotic Normality of the Estimated AUC In thissection we obtain the asymptotic distribution of 120579 = (

1 2

120573 1 2) and hence by the continuous mapping theorem

the asymptotic distribution of the estimated AUC We needto introduce some notation Let

120583119883

119895119896= 119864 (119883

120573(119895+1)(ln119883)

119896)

120583119884

119895119896= 119864 (119884

120573(119895+1)(ln119884)

119896) for 119895 119896 = 0 1 2

A (120579) = (

11988611

0 11988613

11988614

0

0 11988622

11988623

0 11988625

11988631

11988632

11988633

11988634

11988635

11988641

0 11988643

11988644

0

0 11988652

11988653

0 11988655

)

(26)

Then

11988611

=1

1205722

1

11988622

=1

1205722

2

11988613

= 11988631

= minus1205821

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))119895

119895120583119884

1198951

11988623

= 11988632

= minus1

radic1199011205822

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198951

11988625

= 11988652

= minus

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198950

11988633

= 1205821120583119883

02+

1

radic1199011205822120583119884

02+

1 + 119901

radic1199011205732

minus 1205821

(1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198952

minus1

radic1199011205822

(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))

119895(1 minus 120582

2(1 + 119894))

119895

120583119884

1198952

11988634

= 11988643

= 120583119883

01minus (1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198951

11988635

= 11988653

=1

radic119901120583119884

01minus

1

radic119901(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205822

(1 + 119894))

119895(minus1205822

(1 + 119894))119895

120583119884

1198951

11988644

=1

1205822

1

+ (1205721

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205821)119895

120583119883

119895+10

11988655

=1

1205822

2

+ (1205722

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205822)119895

120583119884

119895+10

(27)

6 ISRN Probability and Statistics

Estimated AUC

Freq

uenc

y

05 06 07 08

0

100

200

300

(a)

Estimated AUC

Freq

uenc

y

055 060 070 080

0

100

50

150

200

065 075

(b)

Estimated AUC

Freq

uenc

y

050 055 060 065 070 075 080

0

50

100

150

(c)

Estimated AUC

Freq

uenc

y

055 060 065 070 075

0

50

100

150

200

250

(d)

Figure 1 Histograms of the estimatedAUC for (a) (119898 119899) = (50 50) (b) (119898 119899) = (100 100) (c) (119898 119899) = (50 100) and (d) (119898 119899) = (100 200)

We denote 119863 = diag(radic119898 radic119899 radic119898 radic119898 radic119899) Noticethat EW family satisfies all the regularity conditions almosteverywhere Nowwe are ready to state the following theorem

Theorem 6 Suppose (119883 119884) follows the Bi-EWmodel with themodel parameter 120579 If 119898 rarr infin and 119899 rarr infin but 119898119899 rarr 119901for 0 lt 119901 lt infin then one has the following

(a) TheMLE 120579 of 120579 is asympotic normal

119863 (120579 minus 120579) 997888rarr 1198735

(0Aminus1 (120579)) (28)

(b) Part (a) the continuous mapping theorem and Deltamethod imply that

radic119898 (119860119880119862 minus 119860119880119862) 997888rarr 119873 (0 119861) (29)

where

119861 = (120597119860119880119862

120597120579)Aminus1 (120579) (

120597119860119880119862

120597120579)

119879

120597119860119880119862

120597120579= (

120597119860119880119862

1205971205721

120597119860119880119862

1205971205722

0120597119860119880119862

1205971205731

120597119860119880119862

1205971205732

)

(30)

4 A Simulation Study

In this section we present some results based on MonteCarlo simulation to check the performance of the MLE ofparameters and AUCWe consider small to moderate samplesizes 119898 119899 = 50 100 and 200 We set two scale parameters1205821

= 1205822

= 1 The common shape parameter is 120573 = 2The other shape parameters are 120572

1= 2 and 120572

2= 1 2 3 4

respectively The simulation is based on 1000 replicatesFigure 1 illustrates the histograms of the estimated AUC

under the sample sizes (119898 119899) = (50 50) (100 100) (50 100)

(100 200) when the true parameters are 1205721

= 2 1205722

= 4

ISRN Probability and Statistics 7

Table 1 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120573 = 2 120582

1= 1 and 120582

2= 1 and with

different values of 1205722

(119898 119899) MLE 1205722

= 1 1205722

= 2 1205722

= 3 1205722

= 4

(50 50)

1

03721 (16569) 035630 (2211) 03607 (15814) 03995 (1950)2

02556 (11068) 03645 (12601) 04204 (15814) 05547 (21276)120573 00801 (07525) 00675 (06655) 00615 (06679) 00525 (06243)1

00842 (05661) 00756 (05421) 0961 (05442) 00979 (05683)2

00913 (05235) 00805 (05493) 00916 (0560) 0119 (06189)AUC 00309 (00648) 00187 (00734) 00074 (008) 00078 (00648)

(50 100)

1

01872 (16205) 03035 (21506) 02333 (15885) 02205 (19536)2

00823 (04995) 02404 (18192) 02507 (26231) 02896 (21170)120573 00636 (05464) 00293 (05054) 00271 (04935) 00467 (05228)1

00348 (04380) 00985 (04777) 00859 (04421) 00653 (04614)2

00217 (03694) 00751 (04532) 00762 (04510) 00612 (04861)AUC 00072 (0060) 0005 (00591) 00037 (00565) 00058 (00424)

(100 50)

1

01609 (13638) 01689 (15535) 01774 (13754) 01731 (14751)2

01379 (05709) 02031 (16370) 02773 (19342) 03209 (17746)120573 00467 (05281) 00455 (04998) 00320 (04814) 00425 (05079)1

00399 (04085) 00430 (04129) 00602 (03962) 00461 (04191)2

00709 (04078) 00524 (04300) 00824 (04498) 00668 (04889)AUC 00048 (00519) 00031 (00489) 00017 (00469) 00056 (00435)

(100 100)

1

01205 (11180) 01322 (12528) 01396 (12052) 01276 (11109)2

00726 (04195) 01291 (12499) 01842 (21241) 01295 (20528)120573 00367 (04324) 00346 (04317) 00290 (04219) 00323 (04281)1

00318 (03521) 00354 (03687) 00469 (03720) 00369 (03644)2

0029 (03286) 00323 (03739) 00593 (03962) 00452 (03954)AUC 00036 (00412) 0003 (00458) 00018 (00331) 00031 (0040)

(200 200)

1

00668 (06871) 00569 (06666) 00700 (06368) 00602 (06553)2

00404 (02707) 00637 (06581) 00833 (11019) 00965 (07324)120573 00152 (02879) 00135 (02853) 00090 (02791) 00102 (02756)1

00214 (02487) 00215 (02457) 00324 (02399) 00217 (02430)2

00162 (02274) 00222 (02419) 00292 (02562) 00280 (02715)AUC 00021 (00256) 00033 (00267) 00031 (00270) 00013 (00264)

120573 = 2 and 1205821

= 1205822

= 1 Obviously as the sample sizeincreases the histogram tends to be more symmetric bellshaped Therefore the MLE of AUC performs well for themoderate sample size

We report the absolute relative biases (ARB) and thesquare root of mean squared errors (RMSEs) for the estima-tors of 120579 = (120579

1 120579

5) and AUC defined as below

ARB (120579119895) =

1

1000

1000

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

120579119894

119895minus 120579119895

120579119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

RMSE (120579119895) = radic

1

1000

1000

sum

119894=1

(120579119894

119895minus 120579119895)2

119895 = 1 2 3 4 5

(31)

where 120579119894

119895is the estimates of 120579

119895for the 119894th replicate

Recall that

AUC = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821) (32)

The estimator AUC and true value AUC can be computed byRiemann sums

119887

sum

119909=00001

1198650

(119909 2 120573 2) 1198911

(119909 1 120573 1) Δ119909

119887

sum

119909=00001

1198650

(119909 1205722 120573 1205822) 1198911

(119909 1205721 120573 1205821) Δ119909

(33)

respectively Here we choose 119887 = 4 and Δ119909 = 00001The interval [0 4] is divided evenly with each subintervallength 00001 and 119909 takes the value of the right end of eachsubintervalWe have verified that the summation keeps stablefor any larger value 119887 and smaller value Δ119909

Similarly

ARB (AUC) =1

1000

1000

sum

119894=1

1003816100381610038161003816100381610038161003816100381610038161003816

AUC119894minus AUC

AUC

1003816100381610038161003816100381610038161003816100381610038161003816

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 ISRN Probability and Statistics

where the two distribution functions share a common shapeparameter 120573

Theorem3 Suppose that (119883 119884) follows the Bi-EWmodel withthe above distribution functions Then

119860119880119862 =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894 (14)

Proof By Taylor expansion we have

(1 minus 119890minus1205822119909120573

)

1205722

=

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

(1 minus 119890minus1205821119909120573

)

1205721

=

infin

sum

119895=0

(1205721

119895) (minus119890

minus1205821119909120573

)

119895

(15)

Thus

119860119880119862 = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821)

= int

infin

0

infin

sum

119894=0

(1205722

119894) (minus119890

minus1205822119909120573

)

119894

times

infin

sum

119895=1

(1205721

119895) 119895(minus119890

minus1205821119909120573

)

119895minus1

119890minus1205821119909120573

1205821120573119909120573minus1

119889119909

=

infin

sum

119894=0

infin

sum

119895=1

int

infin

0

(1205722

119894) (

1205721

119895) 119895(minus1)

119894+119895minus11205821119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus11205821119895 int

infin

0

119890minus(1205822119894+1205821119895)119909

120573

119889119909120573

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 1205821119895

1205821119895 + 1205822119894

(16)

Remark 4 (i) Note that the exact expression of AUC isindependent of the common parameter 120573 This is similar tothe cases in Bi-WE and Bi-GE models see Kundu and Gupta[24 25]

(ii) When 1205721

= 1205722

= 1 (14) reduces to the AUC for aBi-GE model

(iii) When 1205821

= 1205822 (14) reduces to the AUC for a Bi-WE

model To see (iii) we actually prove the following corollary

Corollary 5 For 1205721

gt 0 and 1205722

gt 0 one hasinfin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119895 + 119894=

1205721

1205721

+ 1205722

(17)

Proof Denote

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119905119894+119895minus1

119904119895119889119905 (18)

Then on one hand we have

119892 (119904) =120597

120597119904int

1

0

infin

sum

119894=0

(1205722

119894) (minus1)

119894119905119894minus1

infin

sum

119895=1

(1205721

119895) (minus1)

119895minus1(119905119904)119895119889119905

=120597

120597119904int

1

0

1

119905(1 minus 119905)1205722 (1 minus (1 minus 119905119904)

1205721)119889119905

= int

1

0

1

119905(1 minus 119905)

1205722(1 minus 119905119904)1205721minus11205721119905 119889119905

= 1205721

int

1

0

(1 minus 119905)1205722(1 minus 119905119904)

1205721minus1119889119905

(19)

On the other hand we have

119892 (119904) =120597

120597119904

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1119904119895 119905119894+119895

119894 + 119895

=

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895119904119895minus1

119905119894+119895

(20)

Let 119904 = 1 then (19) implies that

119892 (1) =1205721

1205721

+ 1205722

(21)

while (20) implies that

119892 (1) =

infin

sum

119894=0

infin

sum

119895=1

(1205722

119894) (

1205721

119895) (minus1)

119894+119895minus1 119895

119894 + 119895 (22)

This completes the proof of the corollary

3 Maximum Likelihood Estimator ofAUC and Its Asymptotic Property underthe Bi-EW Model

To estimate AUC under the Bi-EWmodel we adopt the plugin method That is we first obtain the maximum likelihoodestimator of the model parameters and then plug it into (14)to get the MLE of AUC

31 Maximum Likelihood Estimator of AUC under the Bi-EWModel Let119883

1 1198832 119883

119898and1198841 1198842 119884

119899be independent

random samples from 119883 sim 1198651and 119884 sim 119865

0 respectively

ISRN Probability and Statistics 5

Denote the model parameter vector by 120579 = (1205791 120579

5) =

(1205721 1205722 120573 1205821 1205822) Then the log-likelihood function is

119897 (120579) = 119898 ln1205721

+ 119898 ln 1205821

+ 119898 ln120573 + 119899 ln1205722

+ 119899 ln 1205822

+ 119899 ln120573 + (1205721

minus 1)

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 )

+ (1205722

minus 1)

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119894 )

minus

119898

sum

119894=1

1205821119909120573

119894+ (120573 minus 1)

119898

sum

119894=1

ln119909119894

minus

119899

sum

119895=1

1205822119910120573

119895+ (120573 minus 1)

119899

sum

119895=1

ln119910119894

(23)

We take derivatives with respect to the parameters to get thefollowing score equations

120597119897

1205971205721

=119898

1205721

+

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 ) = 0

120597119897

1205971205722

=119899

1205722

+

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119895 ) = 0

120597119897

1205971205821

=119898

1205821

minus

119898

sum

119894=1

119909120573

119894+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 119909120573

119894

1 minus 119890minus1205821119909120573119894

= 0

120597119897

1205971205822

=119899

1205822

minus

119899

sum

119895=1

119910120573

119895+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 119910120573

119895

1 minus 119890minus1205822119910120573119895

= 0

120597119897

120597120573=

119898 + 119899

120573minus 1205821

119898

sum

119894=1

119909120573

119894ln119909119894minus 1205822

119899

sum

119895=1

119910120573

119895ln119910119895

+

119898

sum

119894=1

ln119909119894+

119899

sum

119895=1

ln119910119895

+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 1205821119909120573

119894ln119909119894

1 minus 119890minus1205821119909120573119894

+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 1205822119910120573

119895ln119910119895

1 minus 119890minus1205822119910120573119895

= 0

(24)

The MLE 120579 of 120579 is the numerical solution of the above scoreequations Plugging 120579 into (14) we obtain the maximumlikelihood estimator of AUC as below

AUC =

infin

sum

119894=0

infin

sum

119895=1

(2

119894) (

1

119895) (minus1)

119894+119895minus1 1119895

1119895 + 2119894

(25)

32 Asymptotic Normality of the Estimated AUC In thissection we obtain the asymptotic distribution of 120579 = (

1 2

120573 1 2) and hence by the continuous mapping theorem

the asymptotic distribution of the estimated AUC We needto introduce some notation Let

120583119883

119895119896= 119864 (119883

120573(119895+1)(ln119883)

119896)

120583119884

119895119896= 119864 (119884

120573(119895+1)(ln119884)

119896) for 119895 119896 = 0 1 2

A (120579) = (

11988611

0 11988613

11988614

0

0 11988622

11988623

0 11988625

11988631

11988632

11988633

11988634

11988635

11988641

0 11988643

11988644

0

0 11988652

11988653

0 11988655

)

(26)

Then

11988611

=1

1205722

1

11988622

=1

1205722

2

11988613

= 11988631

= minus1205821

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))119895

119895120583119884

1198951

11988623

= 11988632

= minus1

radic1199011205822

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198951

11988625

= 11988652

= minus

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198950

11988633

= 1205821120583119883

02+

1

radic1199011205822120583119884

02+

1 + 119901

radic1199011205732

minus 1205821

(1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198952

minus1

radic1199011205822

(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))

119895(1 minus 120582

2(1 + 119894))

119895

120583119884

1198952

11988634

= 11988643

= 120583119883

01minus (1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198951

11988635

= 11988653

=1

radic119901120583119884

01minus

1

radic119901(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205822

(1 + 119894))

119895(minus1205822

(1 + 119894))119895

120583119884

1198951

11988644

=1

1205822

1

+ (1205721

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205821)119895

120583119883

119895+10

11988655

=1

1205822

2

+ (1205722

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205822)119895

120583119884

119895+10

(27)

6 ISRN Probability and Statistics

Estimated AUC

Freq

uenc

y

05 06 07 08

0

100

200

300

(a)

Estimated AUC

Freq

uenc

y

055 060 070 080

0

100

50

150

200

065 075

(b)

Estimated AUC

Freq

uenc

y

050 055 060 065 070 075 080

0

50

100

150

(c)

Estimated AUC

Freq

uenc

y

055 060 065 070 075

0

50

100

150

200

250

(d)

Figure 1 Histograms of the estimatedAUC for (a) (119898 119899) = (50 50) (b) (119898 119899) = (100 100) (c) (119898 119899) = (50 100) and (d) (119898 119899) = (100 200)

We denote 119863 = diag(radic119898 radic119899 radic119898 radic119898 radic119899) Noticethat EW family satisfies all the regularity conditions almosteverywhere Nowwe are ready to state the following theorem

Theorem 6 Suppose (119883 119884) follows the Bi-EWmodel with themodel parameter 120579 If 119898 rarr infin and 119899 rarr infin but 119898119899 rarr 119901for 0 lt 119901 lt infin then one has the following

(a) TheMLE 120579 of 120579 is asympotic normal

119863 (120579 minus 120579) 997888rarr 1198735

(0Aminus1 (120579)) (28)

(b) Part (a) the continuous mapping theorem and Deltamethod imply that

radic119898 (119860119880119862 minus 119860119880119862) 997888rarr 119873 (0 119861) (29)

where

119861 = (120597119860119880119862

120597120579)Aminus1 (120579) (

120597119860119880119862

120597120579)

119879

120597119860119880119862

120597120579= (

120597119860119880119862

1205971205721

120597119860119880119862

1205971205722

0120597119860119880119862

1205971205731

120597119860119880119862

1205971205732

)

(30)

4 A Simulation Study

In this section we present some results based on MonteCarlo simulation to check the performance of the MLE ofparameters and AUCWe consider small to moderate samplesizes 119898 119899 = 50 100 and 200 We set two scale parameters1205821

= 1205822

= 1 The common shape parameter is 120573 = 2The other shape parameters are 120572

1= 2 and 120572

2= 1 2 3 4

respectively The simulation is based on 1000 replicatesFigure 1 illustrates the histograms of the estimated AUC

under the sample sizes (119898 119899) = (50 50) (100 100) (50 100)

(100 200) when the true parameters are 1205721

= 2 1205722

= 4

ISRN Probability and Statistics 7

Table 1 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120573 = 2 120582

1= 1 and 120582

2= 1 and with

different values of 1205722

(119898 119899) MLE 1205722

= 1 1205722

= 2 1205722

= 3 1205722

= 4

(50 50)

1

03721 (16569) 035630 (2211) 03607 (15814) 03995 (1950)2

02556 (11068) 03645 (12601) 04204 (15814) 05547 (21276)120573 00801 (07525) 00675 (06655) 00615 (06679) 00525 (06243)1

00842 (05661) 00756 (05421) 0961 (05442) 00979 (05683)2

00913 (05235) 00805 (05493) 00916 (0560) 0119 (06189)AUC 00309 (00648) 00187 (00734) 00074 (008) 00078 (00648)

(50 100)

1

01872 (16205) 03035 (21506) 02333 (15885) 02205 (19536)2

00823 (04995) 02404 (18192) 02507 (26231) 02896 (21170)120573 00636 (05464) 00293 (05054) 00271 (04935) 00467 (05228)1

00348 (04380) 00985 (04777) 00859 (04421) 00653 (04614)2

00217 (03694) 00751 (04532) 00762 (04510) 00612 (04861)AUC 00072 (0060) 0005 (00591) 00037 (00565) 00058 (00424)

(100 50)

1

01609 (13638) 01689 (15535) 01774 (13754) 01731 (14751)2

01379 (05709) 02031 (16370) 02773 (19342) 03209 (17746)120573 00467 (05281) 00455 (04998) 00320 (04814) 00425 (05079)1

00399 (04085) 00430 (04129) 00602 (03962) 00461 (04191)2

00709 (04078) 00524 (04300) 00824 (04498) 00668 (04889)AUC 00048 (00519) 00031 (00489) 00017 (00469) 00056 (00435)

(100 100)

1

01205 (11180) 01322 (12528) 01396 (12052) 01276 (11109)2

00726 (04195) 01291 (12499) 01842 (21241) 01295 (20528)120573 00367 (04324) 00346 (04317) 00290 (04219) 00323 (04281)1

00318 (03521) 00354 (03687) 00469 (03720) 00369 (03644)2

0029 (03286) 00323 (03739) 00593 (03962) 00452 (03954)AUC 00036 (00412) 0003 (00458) 00018 (00331) 00031 (0040)

(200 200)

1

00668 (06871) 00569 (06666) 00700 (06368) 00602 (06553)2

00404 (02707) 00637 (06581) 00833 (11019) 00965 (07324)120573 00152 (02879) 00135 (02853) 00090 (02791) 00102 (02756)1

00214 (02487) 00215 (02457) 00324 (02399) 00217 (02430)2

00162 (02274) 00222 (02419) 00292 (02562) 00280 (02715)AUC 00021 (00256) 00033 (00267) 00031 (00270) 00013 (00264)

120573 = 2 and 1205821

= 1205822

= 1 Obviously as the sample sizeincreases the histogram tends to be more symmetric bellshaped Therefore the MLE of AUC performs well for themoderate sample size

We report the absolute relative biases (ARB) and thesquare root of mean squared errors (RMSEs) for the estima-tors of 120579 = (120579

1 120579

5) and AUC defined as below

ARB (120579119895) =

1

1000

1000

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

120579119894

119895minus 120579119895

120579119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

RMSE (120579119895) = radic

1

1000

1000

sum

119894=1

(120579119894

119895minus 120579119895)2

119895 = 1 2 3 4 5

(31)

where 120579119894

119895is the estimates of 120579

119895for the 119894th replicate

Recall that

AUC = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821) (32)

The estimator AUC and true value AUC can be computed byRiemann sums

119887

sum

119909=00001

1198650

(119909 2 120573 2) 1198911

(119909 1 120573 1) Δ119909

119887

sum

119909=00001

1198650

(119909 1205722 120573 1205822) 1198911

(119909 1205721 120573 1205821) Δ119909

(33)

respectively Here we choose 119887 = 4 and Δ119909 = 00001The interval [0 4] is divided evenly with each subintervallength 00001 and 119909 takes the value of the right end of eachsubintervalWe have verified that the summation keeps stablefor any larger value 119887 and smaller value Δ119909

Similarly

ARB (AUC) =1

1000

1000

sum

119894=1

1003816100381610038161003816100381610038161003816100381610038161003816

AUC119894minus AUC

AUC

1003816100381610038161003816100381610038161003816100381610038161003816

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 5

Denote the model parameter vector by 120579 = (1205791 120579

5) =

(1205721 1205722 120573 1205821 1205822) Then the log-likelihood function is

119897 (120579) = 119898 ln1205721

+ 119898 ln 1205821

+ 119898 ln120573 + 119899 ln1205722

+ 119899 ln 1205822

+ 119899 ln120573 + (1205721

minus 1)

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 )

+ (1205722

minus 1)

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119894 )

minus

119898

sum

119894=1

1205821119909120573

119894+ (120573 minus 1)

119898

sum

119894=1

ln119909119894

minus

119899

sum

119895=1

1205822119910120573

119895+ (120573 minus 1)

119899

sum

119895=1

ln119910119894

(23)

We take derivatives with respect to the parameters to get thefollowing score equations

120597119897

1205971205721

=119898

1205721

+

119898

sum

119894=1

ln(1 minus 119890minus1205821119909120573119894 ) = 0

120597119897

1205971205722

=119899

1205722

+

119899

sum

119895=1

ln(1 minus 119890minus1205822119910120573119895 ) = 0

120597119897

1205971205821

=119898

1205821

minus

119898

sum

119894=1

119909120573

119894+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 119909120573

119894

1 minus 119890minus1205821119909120573119894

= 0

120597119897

1205971205822

=119899

1205822

minus

119899

sum

119895=1

119910120573

119895+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 119910120573

119895

1 minus 119890minus1205822119910120573119895

= 0

120597119897

120597120573=

119898 + 119899

120573minus 1205821

119898

sum

119894=1

119909120573

119894ln119909119894minus 1205822

119899

sum

119895=1

119910120573

119895ln119910119895

+

119898

sum

119894=1

ln119909119894+

119899

sum

119895=1

ln119910119895

+ (1205721

minus 1)

119898

sum

119894=1

119890minus1205821119909120573119894 1205821119909120573

119894ln119909119894

1 minus 119890minus1205821119909120573119894

+ (1205722

minus 1)

119899

sum

119895=1

119890minus1205822119910120573119895 1205822119910120573

119895ln119910119895

1 minus 119890minus1205822119910120573119895

= 0

(24)

The MLE 120579 of 120579 is the numerical solution of the above scoreequations Plugging 120579 into (14) we obtain the maximumlikelihood estimator of AUC as below

AUC =

infin

sum

119894=0

infin

sum

119895=1

(2

119894) (

1

119895) (minus1)

119894+119895minus1 1119895

1119895 + 2119894

(25)

32 Asymptotic Normality of the Estimated AUC In thissection we obtain the asymptotic distribution of 120579 = (

1 2

120573 1 2) and hence by the continuous mapping theorem

the asymptotic distribution of the estimated AUC We needto introduce some notation Let

120583119883

119895119896= 119864 (119883

120573(119895+1)(ln119883)

119896)

120583119884

119895119896= 119864 (119884

120573(119895+1)(ln119884)

119896) for 119895 119896 = 0 1 2

A (120579) = (

11988611

0 11988613

11988614

0

0 11988622

11988623

0 11988625

11988631

11988632

11988633

11988634

11988635

11988641

0 11988643

11988644

0

0 11988652

11988653

0 11988655

)

(26)

Then

11988611

=1

1205722

1

11988622

=1

1205722

2

11988613

= 11988631

= minus1205821

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))119895

119895120583119884

1198951

11988623

= 11988632

= minus1

radic1199011205822

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198951

11988625

= 11988652

= minus

infin

sum

119894=0

infin

sum

119895=0

(minus1205822

(1 + 119894))119895

119895120583119884

1198950

11988633

= 1205821120583119883

02+

1

radic1199011205822120583119884

02+

1 + 119901

radic1199011205732

minus 1205821

(1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198952

minus1

radic1199011205822

(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(minus1205821

(1 + 119894))

119895(1 minus 120582

2(1 + 119894))

119895

120583119884

1198952

11988634

= 11988643

= 120583119883

01minus (1205721

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205821

(1 + 119894))

119895(minus1205821

(1 + 119894))119895

120583119883

1198951

11988635

= 11988653

=1

radic119901120583119884

01minus

1

radic119901(1205722

minus 1)

times

infin

sum

119894=0

infin

sum

119895=0

(1 minus 1205822

(1 + 119894))

119895(minus1205822

(1 + 119894))119895

120583119884

1198951

11988644

=1

1205822

1

+ (1205721

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205821)119895

120583119883

119895+10

11988655

=1

1205822

2

+ (1205722

minus 1)

infin

sum

119894=0

infin

sum

119895=0

(119894 + 1)119895+1

119895(minus1205822)119895

120583119884

119895+10

(27)

6 ISRN Probability and Statistics

Estimated AUC

Freq

uenc

y

05 06 07 08

0

100

200

300

(a)

Estimated AUC

Freq

uenc

y

055 060 070 080

0

100

50

150

200

065 075

(b)

Estimated AUC

Freq

uenc

y

050 055 060 065 070 075 080

0

50

100

150

(c)

Estimated AUC

Freq

uenc

y

055 060 065 070 075

0

50

100

150

200

250

(d)

Figure 1 Histograms of the estimatedAUC for (a) (119898 119899) = (50 50) (b) (119898 119899) = (100 100) (c) (119898 119899) = (50 100) and (d) (119898 119899) = (100 200)

We denote 119863 = diag(radic119898 radic119899 radic119898 radic119898 radic119899) Noticethat EW family satisfies all the regularity conditions almosteverywhere Nowwe are ready to state the following theorem

Theorem 6 Suppose (119883 119884) follows the Bi-EWmodel with themodel parameter 120579 If 119898 rarr infin and 119899 rarr infin but 119898119899 rarr 119901for 0 lt 119901 lt infin then one has the following

(a) TheMLE 120579 of 120579 is asympotic normal

119863 (120579 minus 120579) 997888rarr 1198735

(0Aminus1 (120579)) (28)

(b) Part (a) the continuous mapping theorem and Deltamethod imply that

radic119898 (119860119880119862 minus 119860119880119862) 997888rarr 119873 (0 119861) (29)

where

119861 = (120597119860119880119862

120597120579)Aminus1 (120579) (

120597119860119880119862

120597120579)

119879

120597119860119880119862

120597120579= (

120597119860119880119862

1205971205721

120597119860119880119862

1205971205722

0120597119860119880119862

1205971205731

120597119860119880119862

1205971205732

)

(30)

4 A Simulation Study

In this section we present some results based on MonteCarlo simulation to check the performance of the MLE ofparameters and AUCWe consider small to moderate samplesizes 119898 119899 = 50 100 and 200 We set two scale parameters1205821

= 1205822

= 1 The common shape parameter is 120573 = 2The other shape parameters are 120572

1= 2 and 120572

2= 1 2 3 4

respectively The simulation is based on 1000 replicatesFigure 1 illustrates the histograms of the estimated AUC

under the sample sizes (119898 119899) = (50 50) (100 100) (50 100)

(100 200) when the true parameters are 1205721

= 2 1205722

= 4

ISRN Probability and Statistics 7

Table 1 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120573 = 2 120582

1= 1 and 120582

2= 1 and with

different values of 1205722

(119898 119899) MLE 1205722

= 1 1205722

= 2 1205722

= 3 1205722

= 4

(50 50)

1

03721 (16569) 035630 (2211) 03607 (15814) 03995 (1950)2

02556 (11068) 03645 (12601) 04204 (15814) 05547 (21276)120573 00801 (07525) 00675 (06655) 00615 (06679) 00525 (06243)1

00842 (05661) 00756 (05421) 0961 (05442) 00979 (05683)2

00913 (05235) 00805 (05493) 00916 (0560) 0119 (06189)AUC 00309 (00648) 00187 (00734) 00074 (008) 00078 (00648)

(50 100)

1

01872 (16205) 03035 (21506) 02333 (15885) 02205 (19536)2

00823 (04995) 02404 (18192) 02507 (26231) 02896 (21170)120573 00636 (05464) 00293 (05054) 00271 (04935) 00467 (05228)1

00348 (04380) 00985 (04777) 00859 (04421) 00653 (04614)2

00217 (03694) 00751 (04532) 00762 (04510) 00612 (04861)AUC 00072 (0060) 0005 (00591) 00037 (00565) 00058 (00424)

(100 50)

1

01609 (13638) 01689 (15535) 01774 (13754) 01731 (14751)2

01379 (05709) 02031 (16370) 02773 (19342) 03209 (17746)120573 00467 (05281) 00455 (04998) 00320 (04814) 00425 (05079)1

00399 (04085) 00430 (04129) 00602 (03962) 00461 (04191)2

00709 (04078) 00524 (04300) 00824 (04498) 00668 (04889)AUC 00048 (00519) 00031 (00489) 00017 (00469) 00056 (00435)

(100 100)

1

01205 (11180) 01322 (12528) 01396 (12052) 01276 (11109)2

00726 (04195) 01291 (12499) 01842 (21241) 01295 (20528)120573 00367 (04324) 00346 (04317) 00290 (04219) 00323 (04281)1

00318 (03521) 00354 (03687) 00469 (03720) 00369 (03644)2

0029 (03286) 00323 (03739) 00593 (03962) 00452 (03954)AUC 00036 (00412) 0003 (00458) 00018 (00331) 00031 (0040)

(200 200)

1

00668 (06871) 00569 (06666) 00700 (06368) 00602 (06553)2

00404 (02707) 00637 (06581) 00833 (11019) 00965 (07324)120573 00152 (02879) 00135 (02853) 00090 (02791) 00102 (02756)1

00214 (02487) 00215 (02457) 00324 (02399) 00217 (02430)2

00162 (02274) 00222 (02419) 00292 (02562) 00280 (02715)AUC 00021 (00256) 00033 (00267) 00031 (00270) 00013 (00264)

120573 = 2 and 1205821

= 1205822

= 1 Obviously as the sample sizeincreases the histogram tends to be more symmetric bellshaped Therefore the MLE of AUC performs well for themoderate sample size

We report the absolute relative biases (ARB) and thesquare root of mean squared errors (RMSEs) for the estima-tors of 120579 = (120579

1 120579

5) and AUC defined as below

ARB (120579119895) =

1

1000

1000

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

120579119894

119895minus 120579119895

120579119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

RMSE (120579119895) = radic

1

1000

1000

sum

119894=1

(120579119894

119895minus 120579119895)2

119895 = 1 2 3 4 5

(31)

where 120579119894

119895is the estimates of 120579

119895for the 119894th replicate

Recall that

AUC = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821) (32)

The estimator AUC and true value AUC can be computed byRiemann sums

119887

sum

119909=00001

1198650

(119909 2 120573 2) 1198911

(119909 1 120573 1) Δ119909

119887

sum

119909=00001

1198650

(119909 1205722 120573 1205822) 1198911

(119909 1205721 120573 1205821) Δ119909

(33)

respectively Here we choose 119887 = 4 and Δ119909 = 00001The interval [0 4] is divided evenly with each subintervallength 00001 and 119909 takes the value of the right end of eachsubintervalWe have verified that the summation keeps stablefor any larger value 119887 and smaller value Δ119909

Similarly

ARB (AUC) =1

1000

1000

sum

119894=1

1003816100381610038161003816100381610038161003816100381610038161003816

AUC119894minus AUC

AUC

1003816100381610038161003816100381610038161003816100381610038161003816

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Probability and Statistics

Estimated AUC

Freq

uenc

y

05 06 07 08

0

100

200

300

(a)

Estimated AUC

Freq

uenc

y

055 060 070 080

0

100

50

150

200

065 075

(b)

Estimated AUC

Freq

uenc

y

050 055 060 065 070 075 080

0

50

100

150

(c)

Estimated AUC

Freq

uenc

y

055 060 065 070 075

0

50

100

150

200

250

(d)

Figure 1 Histograms of the estimatedAUC for (a) (119898 119899) = (50 50) (b) (119898 119899) = (100 100) (c) (119898 119899) = (50 100) and (d) (119898 119899) = (100 200)

We denote 119863 = diag(radic119898 radic119899 radic119898 radic119898 radic119899) Noticethat EW family satisfies all the regularity conditions almosteverywhere Nowwe are ready to state the following theorem

Theorem 6 Suppose (119883 119884) follows the Bi-EWmodel with themodel parameter 120579 If 119898 rarr infin and 119899 rarr infin but 119898119899 rarr 119901for 0 lt 119901 lt infin then one has the following

(a) TheMLE 120579 of 120579 is asympotic normal

119863 (120579 minus 120579) 997888rarr 1198735

(0Aminus1 (120579)) (28)

(b) Part (a) the continuous mapping theorem and Deltamethod imply that

radic119898 (119860119880119862 minus 119860119880119862) 997888rarr 119873 (0 119861) (29)

where

119861 = (120597119860119880119862

120597120579)Aminus1 (120579) (

120597119860119880119862

120597120579)

119879

120597119860119880119862

120597120579= (

120597119860119880119862

1205971205721

120597119860119880119862

1205971205722

0120597119860119880119862

1205971205731

120597119860119880119862

1205971205732

)

(30)

4 A Simulation Study

In this section we present some results based on MonteCarlo simulation to check the performance of the MLE ofparameters and AUCWe consider small to moderate samplesizes 119898 119899 = 50 100 and 200 We set two scale parameters1205821

= 1205822

= 1 The common shape parameter is 120573 = 2The other shape parameters are 120572

1= 2 and 120572

2= 1 2 3 4

respectively The simulation is based on 1000 replicatesFigure 1 illustrates the histograms of the estimated AUC

under the sample sizes (119898 119899) = (50 50) (100 100) (50 100)

(100 200) when the true parameters are 1205721

= 2 1205722

= 4

ISRN Probability and Statistics 7

Table 1 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120573 = 2 120582

1= 1 and 120582

2= 1 and with

different values of 1205722

(119898 119899) MLE 1205722

= 1 1205722

= 2 1205722

= 3 1205722

= 4

(50 50)

1

03721 (16569) 035630 (2211) 03607 (15814) 03995 (1950)2

02556 (11068) 03645 (12601) 04204 (15814) 05547 (21276)120573 00801 (07525) 00675 (06655) 00615 (06679) 00525 (06243)1

00842 (05661) 00756 (05421) 0961 (05442) 00979 (05683)2

00913 (05235) 00805 (05493) 00916 (0560) 0119 (06189)AUC 00309 (00648) 00187 (00734) 00074 (008) 00078 (00648)

(50 100)

1

01872 (16205) 03035 (21506) 02333 (15885) 02205 (19536)2

00823 (04995) 02404 (18192) 02507 (26231) 02896 (21170)120573 00636 (05464) 00293 (05054) 00271 (04935) 00467 (05228)1

00348 (04380) 00985 (04777) 00859 (04421) 00653 (04614)2

00217 (03694) 00751 (04532) 00762 (04510) 00612 (04861)AUC 00072 (0060) 0005 (00591) 00037 (00565) 00058 (00424)

(100 50)

1

01609 (13638) 01689 (15535) 01774 (13754) 01731 (14751)2

01379 (05709) 02031 (16370) 02773 (19342) 03209 (17746)120573 00467 (05281) 00455 (04998) 00320 (04814) 00425 (05079)1

00399 (04085) 00430 (04129) 00602 (03962) 00461 (04191)2

00709 (04078) 00524 (04300) 00824 (04498) 00668 (04889)AUC 00048 (00519) 00031 (00489) 00017 (00469) 00056 (00435)

(100 100)

1

01205 (11180) 01322 (12528) 01396 (12052) 01276 (11109)2

00726 (04195) 01291 (12499) 01842 (21241) 01295 (20528)120573 00367 (04324) 00346 (04317) 00290 (04219) 00323 (04281)1

00318 (03521) 00354 (03687) 00469 (03720) 00369 (03644)2

0029 (03286) 00323 (03739) 00593 (03962) 00452 (03954)AUC 00036 (00412) 0003 (00458) 00018 (00331) 00031 (0040)

(200 200)

1

00668 (06871) 00569 (06666) 00700 (06368) 00602 (06553)2

00404 (02707) 00637 (06581) 00833 (11019) 00965 (07324)120573 00152 (02879) 00135 (02853) 00090 (02791) 00102 (02756)1

00214 (02487) 00215 (02457) 00324 (02399) 00217 (02430)2

00162 (02274) 00222 (02419) 00292 (02562) 00280 (02715)AUC 00021 (00256) 00033 (00267) 00031 (00270) 00013 (00264)

120573 = 2 and 1205821

= 1205822

= 1 Obviously as the sample sizeincreases the histogram tends to be more symmetric bellshaped Therefore the MLE of AUC performs well for themoderate sample size

We report the absolute relative biases (ARB) and thesquare root of mean squared errors (RMSEs) for the estima-tors of 120579 = (120579

1 120579

5) and AUC defined as below

ARB (120579119895) =

1

1000

1000

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

120579119894

119895minus 120579119895

120579119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

RMSE (120579119895) = radic

1

1000

1000

sum

119894=1

(120579119894

119895minus 120579119895)2

119895 = 1 2 3 4 5

(31)

where 120579119894

119895is the estimates of 120579

119895for the 119894th replicate

Recall that

AUC = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821) (32)

The estimator AUC and true value AUC can be computed byRiemann sums

119887

sum

119909=00001

1198650

(119909 2 120573 2) 1198911

(119909 1 120573 1) Δ119909

119887

sum

119909=00001

1198650

(119909 1205722 120573 1205822) 1198911

(119909 1205721 120573 1205821) Δ119909

(33)

respectively Here we choose 119887 = 4 and Δ119909 = 00001The interval [0 4] is divided evenly with each subintervallength 00001 and 119909 takes the value of the right end of eachsubintervalWe have verified that the summation keeps stablefor any larger value 119887 and smaller value Δ119909

Similarly

ARB (AUC) =1

1000

1000

sum

119894=1

1003816100381610038161003816100381610038161003816100381610038161003816

AUC119894minus AUC

AUC

1003816100381610038161003816100381610038161003816100381610038161003816

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 7

Table 1 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120573 = 2 120582

1= 1 and 120582

2= 1 and with

different values of 1205722

(119898 119899) MLE 1205722

= 1 1205722

= 2 1205722

= 3 1205722

= 4

(50 50)

1

03721 (16569) 035630 (2211) 03607 (15814) 03995 (1950)2

02556 (11068) 03645 (12601) 04204 (15814) 05547 (21276)120573 00801 (07525) 00675 (06655) 00615 (06679) 00525 (06243)1

00842 (05661) 00756 (05421) 0961 (05442) 00979 (05683)2

00913 (05235) 00805 (05493) 00916 (0560) 0119 (06189)AUC 00309 (00648) 00187 (00734) 00074 (008) 00078 (00648)

(50 100)

1

01872 (16205) 03035 (21506) 02333 (15885) 02205 (19536)2

00823 (04995) 02404 (18192) 02507 (26231) 02896 (21170)120573 00636 (05464) 00293 (05054) 00271 (04935) 00467 (05228)1

00348 (04380) 00985 (04777) 00859 (04421) 00653 (04614)2

00217 (03694) 00751 (04532) 00762 (04510) 00612 (04861)AUC 00072 (0060) 0005 (00591) 00037 (00565) 00058 (00424)

(100 50)

1

01609 (13638) 01689 (15535) 01774 (13754) 01731 (14751)2

01379 (05709) 02031 (16370) 02773 (19342) 03209 (17746)120573 00467 (05281) 00455 (04998) 00320 (04814) 00425 (05079)1

00399 (04085) 00430 (04129) 00602 (03962) 00461 (04191)2

00709 (04078) 00524 (04300) 00824 (04498) 00668 (04889)AUC 00048 (00519) 00031 (00489) 00017 (00469) 00056 (00435)

(100 100)

1

01205 (11180) 01322 (12528) 01396 (12052) 01276 (11109)2

00726 (04195) 01291 (12499) 01842 (21241) 01295 (20528)120573 00367 (04324) 00346 (04317) 00290 (04219) 00323 (04281)1

00318 (03521) 00354 (03687) 00469 (03720) 00369 (03644)2

0029 (03286) 00323 (03739) 00593 (03962) 00452 (03954)AUC 00036 (00412) 0003 (00458) 00018 (00331) 00031 (0040)

(200 200)

1

00668 (06871) 00569 (06666) 00700 (06368) 00602 (06553)2

00404 (02707) 00637 (06581) 00833 (11019) 00965 (07324)120573 00152 (02879) 00135 (02853) 00090 (02791) 00102 (02756)1

00214 (02487) 00215 (02457) 00324 (02399) 00217 (02430)2

00162 (02274) 00222 (02419) 00292 (02562) 00280 (02715)AUC 00021 (00256) 00033 (00267) 00031 (00270) 00013 (00264)

120573 = 2 and 1205821

= 1205822

= 1 Obviously as the sample sizeincreases the histogram tends to be more symmetric bellshaped Therefore the MLE of AUC performs well for themoderate sample size

We report the absolute relative biases (ARB) and thesquare root of mean squared errors (RMSEs) for the estima-tors of 120579 = (120579

1 120579

5) and AUC defined as below

ARB (120579119895) =

1

1000

1000

sum

119894=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

120579119894

119895minus 120579119895

120579119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

RMSE (120579119895) = radic

1

1000

1000

sum

119894=1

(120579119894

119895minus 120579119895)2

119895 = 1 2 3 4 5

(31)

where 120579119894

119895is the estimates of 120579

119895for the 119894th replicate

Recall that

AUC = int

infin

0

1198650

(119909 1205722 120573 1205822) 1198891198651

(119909 1205721 120573 1205821) (32)

The estimator AUC and true value AUC can be computed byRiemann sums

119887

sum

119909=00001

1198650

(119909 2 120573 2) 1198911

(119909 1 120573 1) Δ119909

119887

sum

119909=00001

1198650

(119909 1205722 120573 1205822) 1198911

(119909 1205721 120573 1205821) Δ119909

(33)

respectively Here we choose 119887 = 4 and Δ119909 = 00001The interval [0 4] is divided evenly with each subintervallength 00001 and 119909 takes the value of the right end of eachsubintervalWe have verified that the summation keeps stablefor any larger value 119887 and smaller value Δ119909

Similarly

ARB (AUC) =1

1000

1000

sum

119894=1

1003816100381610038161003816100381610038161003816100381610038161003816

AUC119894minus AUC

AUC

1003816100381610038161003816100381610038161003816100381610038161003816

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 ISRN Probability and Statistics

Table 2 ARB and RMSE of five estimated parameters (1 2 120573 1 and

2) and AUC when 120572

1= 2 120572

2= 3 120582

1= 1 and 120582

2= 1 and with

different values of 120573

(119898 119899) MLE 120573 = 2 120573 = 25 120573 = 3

(50 50)

1

01925 (17453) 01646 (17159) 01863 (17626)2

04685 (2245) 03917 (19021) 04641 (2348)120573 00554 (06402) 00912 (07801) 00558 (09117)1

01044 (05624) 00700 (05442) 00800 (05353)2

01048 (05779) 00746 (05624) 00939 (05719)AUC 00067 (00556) 00026 (00557) 00036 (00529)

(50 80)

1

02538 (18319) 02472 (20245) 02775 (20910)2

02818 (20819) 02691 (2245) 03046 (23255)120573 00401 (05415) 00557 (06907) 00475 (08629)1

00825 (04706) 00635 (04821) 00815 (04803)2

00724 (04790) 00489 (04931) 00741 (04952)AUC 00045 (00490) 00015 (00510) 00008 (0050)

(80 50)

1

01846 (16341) 02311 (17011) 02874 (17425)2

02515 (22201) 03733 (22412) 03969 (1956)120573 00501 (04921) 00414 (0701) 00339 (08101)1

00380 (04365) 00665 (04577) 00905 (04761)2

00443 (04681) 00952 (05278) 00962 (05221)AUC 00003 (00479) 00013 (00479) 00055 (00489)

(80 80)

1

0162 (13518) 01531 (13652) 01594 (13561)2

02096 (18697) 02047 (39241) 01655 (2183)120573 00372 (04715) 00438 (05934) 00458 (07095)1

00464 (04011) 00404 (03996) 00433 (04048)2

00488 (04211) 00417 (04334) 00291 (04084)AUC 0002 (00412) 00023 (00424) 0003 (00435)

RMSE (AUC) = radic1

1000

1000

sum

119894=1

(AUC119894minus AUC)

2

(34)

where AUC119894is the estimate of AUC for the 119894th replicate

Tables 1 and 2 report the ARB and the RMSE for theestimators of 120572

1 1205722 120573 1205821 1205822 and the AUC Table 1 reports

the sensitivity analysis against 1205722and sample sizes In all

cases the two scale parameters have much smaller ARB andRMSE compared with shape parameters The MLE of AUCbehaves well For two different shape parameters it is alwaysthe commonparameter120573which behaves better than the othertwo shape parameters 120572

1and 120572

2as expected For two shape

parameters 1205721and 120572

2 when 120572

2gt 1205721 1205721always behaves

better than 1205722 Furthermore as expected the ARB and the

RMSE decrease as the sample size increases Table 2 reportsthe sensitivity analysis against the common shape parameter120573 The results show the robustness of the MLEs for smallsample sizes

Our second simulations are to check the performanceof MLE as the common shape parameter 120573 increases Wechoose the following sample sizes (119898 119899) = (50 50) (80 80)(50 80) (80 50) The two scale parameters are still set to 1and 120572

1= 2 120572

2= 3 and 120573 takes three values 2 25 and 3

The results are reported in Table 2 Once again one notices

that AUC has very small ARB and RMSE Under all settings1205722has larger ARB and RMSE than that of 120572

1when 120572

2gt 1205721

When 120573 increases we do not find a noticeable change of ARBand RMSE of all parameters

5 Conclusions

In summary this paper considers an Exponentiated Weibullmodel with two shape parameters and one scale parameterFirstly we derive a general moment formula which extendsmany results in the literature Secondly we obtain thetheoretical AUC formula for a Bi-Exponentiated Weibullmodel which shares a common shape parameter betweenthe two groups We have noticed that the formula of AUCis independent of the common shape parameter which isconsistent with existing known results for simpler modelssuch as Bi-generalized exponential and Bi-Weibull Thirdlywe derive the maximum likelihood estimator of AUC for theBi-EWmodel with a common shape parameter and show theasymptotic normality of the estimators Finally we conducta simulation study to illustrate the performance of ourestimator for small to moderate sample sizes The simulationresults show that theMLE of AUC can behave very well undermoderate sample sizes although the performance of the MLEof some shape parameters is somewhat unsatisfactory but stillacceptable

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 9

References

[1] J A Hanley ldquoReceiver operating characteristic (ROC)method-ology the state of the artrdquo Critical Reviews in DiagnosticImaging vol 29 no 3 pp 307ndash335 1989

[2] C B Begg ldquoAdvances in statistical methodology for diagnosticmedicine in the 1980rsquosrdquo Statistics in Medicine vol 10 no 12 pp1887ndash1895 1991

[3] M S Pepe The Statistical Evaluation of Medical Tests forClassification and Prediction vol 28 ofOxford Statistical ScienceSeries Oxford University Press Oxford 2003

[4] D M Green and J A Swets Singal Detection Theory andPsychophysics Wiley and Sons New York NY USA 1966

[5] H Fushing and B W Turnbull ldquoNonparametric and semi-parametric estimation of the receiver operating characteristiccurverdquoThe Annals of Statistics vol 24 no 1 pp 25ndash40 1996

[6] C EMetz B A Herman and J H Shen ldquoMaximum likelihoodestimation of receiver operating characteristic curve fromcontinuous distributed datardquo Statistics in Medicine vol 17 pp1033ndash1053 1998

[7] K H Zou and W J Hall ldquoTwo transformation models forestimating an ROC curve derived from continuous datardquoJournal of Applied Statistics vol 5 pp 621ndash631 2000

[8] J Gu S Ghosal and A RoyNon-Parametric Estimation of ROCCurve Institute of Statistics Mimeo Series 2005

[9] M S Pepe and T Cai ldquoThe analysis of placement values forevaluating discriminatory measuresrdquo Biometrics vol 60 no 2pp 528ndash535 2004

[10] X-H Zhou andH Lin ldquoSemi-parametricmaximum likelihoodestimates for ROC curves of continuous-scales testsrdquo Statisticsin Medicine vol 27 no 25 pp 5271ndash5290 2008

[11] D Faraggi B Reiser and E F Schisterman ldquoROC curveanalysis for biomarkers based on pooled assessmentsrdquo Statisticsin Medicine vol 22 no 15 pp 2515ndash2527 2003

[12] A J Simpson and M J Fitter ldquoWhat is the best index ofdetectabilityrdquo Psychological Bulletin vol 80 no 6 pp 481ndash4881973

[13] M H Gail and S B Green ldquoA generalization of the one-sided two-sample Kolmogorov-Smirnov statistic for evaluatingdiagnostic testsrdquo Biometrics vol 32 no 3 pp 561ndash570 1976

[14] W-C Lee andC KHsiao ldquoAlternative summary indices for thereceiver operating characteristic curverdquoEpidemiology vol 7 no6 pp 605ndash611 1996

[15] G Campbell ldquoAdvances in statistical methodology for theevaluation of diagnostic and laboratory testsrdquo Statistics inMedicine vol 13 no 5-7 pp 499ndash508 1994

[16] S Kotz Y Lumelskii and M PenskyThe Stress-Strength Modeland Its Generalizations 2002

[17] A M Awad and M A Hamadan ldquoSome inference results inPr(X lt Y) in the bivariate exponential modelrdquoCommunicationsin Statistics vol 10 no 24 pp 2515ndash2524 1981

[18] W A Woodward and G D Kelley ldquoMinimum variance unbi-ased estimation of P(Y lt X) in the normal caserdquo Technometricsvol 19 pp 95ndash98 1997

[19] K Constantine and M Karson ldquoThe Estimation of P(X gt Y) inGamma caserdquo Communication in Statistics-Computations andSimulations vol 15 pp 65ndash388 1986

[20] K E Ahmad M E Fakhry and Z F Jaheen ldquoEmpirical Bayesestimation of P(Y lt X) and characterizations of Burr-type 119883

modelrdquo Journal of Statistical Planning and Inference vol 64 no2 pp 297ndash308 1997

[21] J G Surles andW J Padgett ldquoInference for P(X gtY) in the Burrtype X modelrdquo Journal of Applied Statistical Sciences vol 7 pp225ndash238 1998

[22] J G Surles andW J Padgett ldquoInference for reliability and stress-strength for a scaled Burr type X distributionrdquo Lifetime DataAnalysis vol 7 no 2 pp 187ndash200 2001

[23] M Z Raqab and D Kundu ldquoComparison of different esti-mation of P(X gt Y) for a Scaled Burr Type X DistributionrdquoCommunications in Statistica-Simulation and Computation vol22 pp 122ndash150 2005

[24] D Kundu and R D Gupta ldquoEstimation of P(X gt Y) forgeneralized exponential distributionrdquoMetrika vol 61 no 3 pp291ndash308 2005

[25] D Kundu and R D Gupta ldquoEstimation of P(X gtY) ForWeibulldistributionrdquo IEEE Transactions on Reliability vol 34 pp 201ndash226 2006

[26] E W Stacy ldquoA generalization of the gamma distributionrdquoAnnals of Mathematical Statistics vol 33 pp 1187ndash1192 1962

[27] R L Prentice ldquoDiscrimination among some parametric mod-elsrdquo Biometrika vol 62 no 3 pp 607ndash614 1975

[28] D J Slymen and P A Lachenbruch ldquoSurvival distributionsarising from two families and generated by transformationsrdquoCommunications in Statistics A vol 13 no 10 pp 1179ndash12011984

[29] D P Gaver and M Acar ldquoAnalytical hazard representations foruse in reliability mortality and simulation studiesrdquo Communi-cations in Statistics B vol 8 no 2 pp 91ndash111 1979

[30] U Hjorth ldquoA reliability distribution with increasing decreas-ing constant and bathtub-shaped failure ratesrdquo Technometricsvol 22 no 1 pp 99ndash107 1980

[31] 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

[32] G SMudholkar andA D Hutson ldquoThe exponentiatedWeibullfamily some properties and a flood data applicationrdquo Commu-nications in Statistics vol 25 no 12 pp 3059ndash3083 1996

[33] A L Khedhairi A Sarhan and L Tadj Estimation of the Gen-eralized RayleighDistribution Parameters King SaudUniversity2007

[34] A Baklizi ldquoInference on Pr(X lt Y) in the two-parameterWeibull model based on recordsrdquo ISRN Probability and Statis-tics vol 2012 Article ID 263612 11 pages 2012

[35] L Qian ldquoThe Fisher information matrix for a three-parameterexponentiated Weibull distribution under type II censoringrdquoStatistical Methodology vol 9 no 3 pp 320ndash329 2012

[36] A Choudhury ldquoA simple derivation of moments of the expo-nentiatedWeibull distributionrdquoMetrika vol 62 no 1 pp 17ndash222005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of