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
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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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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
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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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
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
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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
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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
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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