Discriminating Between the Bivariate Generalized Exponential and Bivariate Weibull distributions - PDF document

DiscriminatingBetweentheBivariateGeneralizedExponentialandBivariateWeibullDistributionsArabinKumarDey&DebasisKunduyzAbstractRecentlyKunduandGupta(\Bivariategeneralizedexponentialdistribution",Jour-nalofMultivariateAnalysis,vol.100,581-593,2009)introducedabivariategener-alizedexponentialdistribution,whosemarginalsaregeneralizedexponentialdistribu-tions.Thebivariategeneralizedexponentialdistributionisasingulardistribution,similarlyasthewellknownMarshall-OlkinbivariateWeibulldistribution.Thetwosingularbivariatedistributionsfunctionshaveverysimilarjointprobabilitydensityfunctions.Inthispaperweconsiderthediscriminationbetweenthetwobivariatedis-tributionfunctions.Thedierenceofthemaximizedlog-likelihoodfunctionsisusedindiscriminatingbetweenthetwodistributionfunctions.Theasymptoticdistributionoftheteststatistichasbeenobtainedanditcanbeusedtocomputetheasymptoticprobabilityofcorrectselection.MonteCarlosimulationsareperformedtostudytheeectivenessoftheproposedmethod.Onedatasethasbeenanalyzedforillustrativepurposes.KeyWordsandPhrases:Likelihoodratiotest;asymptoticdistribution;probabilityofcorrectselection;MonteCarlosimulations;EMalgorithm;maximumlikelihoodestimator.yDepartmentofMathematicsandStatistics,IndianInstituteofTechnologyKanpur,Pin208016,India.zCorrespondingauthor.e-mail:kundu@iitk.ac.in1 IntroductionRecently,thetwo-parametergeneralizedexponential(GE)distributionproposedbyGuptaandKundu(1999)hasreceivedsomeattention.Thetwo-parameterGEdistribution,whichhasoneshapeparameter,andonescaleparameterisapositivelyskeweddistribution.Ithasseveraldesirableproperties,andmanyofitspropertiesareverysimilartothecorrespond-ingpropertiesofthewellknownWeibulldistribution.Forexample,theprobabilitydensityfunctions(PDFs)andthehazardfunctions(HFs)oftheGEdistributionandWeibulldistri-butionareverysimilar,andboththedistributionshavenicecompactdistributionfunctions.Boththedistributionscontainexponentialdistributionasaspecialcase,therefore,theyareextensionsoftheexponentialdistributionsbutindierentmanners.ItisfurtherobservedthattheGEdistributionalsocanbeusedquitesuccessfullyinanalyzingpositivelyskeweddatasets,inplaceofWeibulldistribution,andoftenitisverydiculttodistinguishbetweenthetwo.ForsomerecentdevelopmentsonGEdistribution,andforitsdierentapplications,thereadersarereferredtothereviewarticlebyGuptaandKundu(2007).Theproblemoftestingwhethersomegivenobservationsfollowoneofthetwo(ormore)probabilitydistributionfunctions,isquiteanoldstatisticalproblem.Cox(1961),seealsoCox(1962),rstconsideredthisprobleminhisclassicalpaper,andhealsodiscussedtheeectofchoosingthewrongmodel.Sincethenextensiveworkhasbeendoneindiscriminatingbetweentwoormoredistributionfunctions,seeforexampleAtkinson(1969,1970),BainandEnglehardt(1980),Marshall,MezaandOlkin(2001),DeyandKundu(2009,2010)andthereferencescitedtherein.Inrecenttimesitisobserved,seeGuptaandKundu(2003,2006),thatduetotheclosenessbetweenWeibullandGEdistributions,itisextremelydiculttodiscriminatebetweenthesetwodistributionfunctions.Notethatiftheshapeparameterisone,thetwo2 distributionfunctionsarenotdistinguishable.Forsmallsamplesizestheprobabilityofcorrectselection(PCS)canbequiteloweveniftheshapeparameterisnotveryclosetoone.Interestingly,althoughextensiveworkhasbeendoneindiscriminatingbetweentwoormoreunivariatedistributionfunctions,butnoworkhasbeenfoundindiscriminatingbetweentwobivariatedistributionfunctions.RecentlyKunduandGupta(2009)introducedasingularbivariatedistributionfunctionwhosemarginalsareGEdistributionfunctions,andnameditasthebivariategeneralizedexponentialdistribution(BVGE).Thefour-parameterBVGEdistributionhasseveraldesir-ableproperties,anditcanbeusedquiteeectivelytoanalyzebivariatedatawhenthereareties.Anotherfour-parameterwellknownbivariatesingulardistributionistheMarshall-OlkinbivariateWeibull(MOBW)distribution,whichhasbeenusedquiteeectivelytoanalyzebi-variatedatawhenthereareties,seeforexampleKotz,BalakrishnanandJohnson(2000).TheMOBWdistributionhasWeibullmarginals.Therefore,itisclearthatforcertainrangeofparametervalues,themarginalsoftheBVGEandMOBWwillbeverysimilar.InfactitisobservedthattheshapesofthejointPDFsofBVGEandMOBWalsocanbeverysimilarinnature.InthispaperweconsiderdiscriminatingbetweenBVGEandMOBWdistributions.Wehaveusedthedierenceofthemaximizedlog-likelihoodvaluesindiscriminatingbetweenthetwodistributionfunctions.Theexactdistributionoftheproposedteststatisticisdiculttoobtain,andhenceweobtainitsasymptoticdistribution.Itisobservedthattheasymptoticdistributionoftheteststatisticisnormallydistributedandithasbeenusedtocomputetheprobabilityofcorrectselection(PCS).IncomputingthePCSoneneedstocomputethemisspeciedparameters.Computationofthemisspeciedparametersinvolvessolvingafourdimensionaloptimizationproblem.Wesuggestanapproximation,whichinvolvessolvingaonedimensionaloptimizationproblemonly.Therefore,computationallyitbecomesvery3 ecient.MonteCarlosimulationsareperformedtostudytheeectivenessoftheproposedmethod,anditisobservedthatevenformoderatesamplesizestheasymptoticresultsmatchverywellwiththesimulatedresults.Weperformtheanalysisofadatasetforillustrativepurposes.Restofthepaperisorganizedasfollows.InSection2,webrie\rydiscussabouttheBVGEandMOBWdistributions.ThediscriminationprocedureispresentedinSection3.TheasymptoticdistributionoftheteststatisticsforboththecasesareprovidedinSection4.ThecalculationofthemisspeciedparametersarediscussedinSection5.InSection6,wepresenttheMonteCarlosimulationresultsandtheanalysisofadatasetispresentedinSection7.FinallyweconcludethepaperinSection8.MOBWandBVGEDistributionsInthissectionwewillbrie\rydiscussabouttheMOBWandBVGEdistributions.Wewillbeusingthefollowingnotationsthroughoutthepaper.ItisassumedthattheunivariateWeibulldistributionwiththeshapeparameter�0andthescaleparameter�0hasthefollowingprobabilitydensityfunction(PDF),forx�0;fWE(x;;)=x1ex;(1)thecorrespondingcumulativedistributionfunction(CDF)andsurvivalfunction(SF)areWE(x;;)=1ex;andSWE(x;;)=ex;(2)respectively.FromnowonaWeibulldistributionwiththePDFasgivenin(1)willbedenotedbyWE(;).TheGEdistributionwiththeshapeparameter�0andthescaleparameter�0,hasthePDFfGE(x;;)=ex1ex1:(3)4 ThecorrespondingCDFandSFareGE(x;;)=1ex;andSGE(x;;)=11ex(4)respectively.AGEdistributionwiththePDFgivenin(3)willbedenotedbyGE(;).2.1MOBWDistributionSupposeU0WE(;0),U1WE(;1)andU2WE(;2)andtheyareindependentlydistributed.DeneX1=minfU0;U1gandX2=minfU0;U2g,thenthebivariatevector(X1;X2)hastheMOBWdistributionwithparameters;0;1;2,anditwillbedenotedasMOBW(),where=(;0;1;2).If(X1;X2)MOBW(),thentheirjointSFtakesthefollowingform,forz=maxfx1;x2g,SMOBW(x1;x2;)=P(X1�x1;X2�x2)=P(U1�x1;U2�x2;U0�z)=SWE(x1;;1)SWE(x2;;2)SWE(z;;0):(5)ThejointPDFof(X1;X2)canbewrittenasfMOBW(x1;x2;)=f1W(x1;x2;)if0x1x2f2W(x1;x2;)if0x2x1f0W(x;)if0x1=x2=x;(6)wheref1W(x1;x2;)=fWE(x1;;1)fWE(x2;;0+2)f2W(x1;x2;)=fWE(x1;;0+1)fWE(x2;;2)f0W(x;)=0 0+1+2fWE(x;;0+1+2):NotethatthefunctionfMOBW()maybeconsideredtobeadensityfunctionforMOBWdistribution,ifitisunderstoodthatthersttwotermsaredensitieswithrespecttotwo5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.5 1 1.5 2 2.5 3 3.5 4 (b) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (c) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (d)Figure1:SurfaceplotsoftheabsolutecontinuouspartofthejointPDFofMOBWfor(;1;2;3):(a)(2.0,1.0,1.0,1.0)(b)(5.0,1.0,1.0,1.0)(c)(2.0,2.0,2.0,2.0)(d)(1.0,1.0,1.0,1.0).dimensionalLebesguemeasure,andthethirdtermisadensityfunctionwithrespecttoaonedimensionalLebesguemeasure,seeforexampleBemisetal.(1972).ItisclearthatMOBWdistributionhasanabsolutecontinuouspartonf(x1;x2);0x11;0x21;x1=x2g,andasingularpartonf(x1;x2);0x11;0x21;x1=x2g.ThesurfaceplotoftheabsolutecontinuouspartofthejointPDFhasbeenprovidedinFigure1fordierentparametervalues.ItisimmediatethatthejointPDFofMOBWdistributioncantakevarietyofshapes,therefore,itcanbeusedquiteeectivelyinanalyzingsingularbivariatedata.Thefollowingprobabilitieswillbeusedlaterinderivingtheasymptoticprobabilityof6 correctselection.If(X1;X2)MOBW(),thenp1W=P[X1X2]=10y0fWE(x;;1)fWE(y;;0+2)dxdy=1 0+1+2;(7)p2W=P[X1�X2]=101yfWE(x;;0+1)fWE(y;;2)dxdy=2 0+1+2;(8)p0W=P[X1=X2]=0 0+1+210fWE(z;;0+1+2)dz=0 0+1+2:(9)2.2BVGEDistributionSupposeV0GE(0;),V1GE(1;)andV2GE(2;).DeneY1=maxfV0;V1gandY2=maxfV0;V2g.Thenthebivariaterandomvector(Y1;Y2)issaidtohaveBVGEdistributionwithparameters0;1;2;,anditwillbedenotedbyBVGE(),where=(0;1;2;).ItisimmediateY1GE(0+1;),andY2GE(0+2;).ThejointCDFof(Y1;Y2)canbeexpressedasfollowsforv=minfy1;y2g.BVGE(y1;y2;)=P(Y1y1;Y2y2)=P(V1y1;V2y2;V0v)=(1ey1)1(1ey2)2(1ev)0:(10)Inthiscasealso,thejointCDFofY1andY2canbewrittenasfBVGE(y1;y2;)=f1G(y1;y2)if0y1y2f2G(y1;y2)if0y2y1f0G(y)if0y1=y2=y;(11)wheref1G(y1;y2;)=fGE(y1;0+1;)fGE(y2;2;)7 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.02 0.04 0.06 0.08 0.1 0.12 (b) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 (c) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 (d)Figure2:SurfaceplotsoftheabsolutecontinuouspartofthejointPDFofBVGEfor(1;2;3;):(a)(1.0,1.0,2.0,1.0)(b)(1.0,1.0,1.0,4.0)(c)(5.0,5.0,5.0,1.0)(d)(0.5,0.5,0.5,1.0).f2G(y1;y2;)=fGE(y1;1;)fGE(y2;0+2;)f0G(y;)=0 0+1+2fGE(y;0+1+2;):ItisclearthattheBVGEdistributionhasalsoasingularpartandanabsolutecontinuouspartsimilarlyastheMOBWdistribution.ThesurfaceplotofthejointPDFoftheBVGEisprovidedinFigure2,fordierentparametervalues.ItisclearthattheshapeofthejointPDFofBVGEandMOBWareverysimilar.Thefollowingprobabilitieswillbeneededlater.If(Y1;Y2)BVGE(),thenp1G=P[Y1Y2]=10y0fGE(x;0+1;)fGE(y;2;)dxdy=2 0+1+2;(12)8 p2G=P[Y1�Y2]=101yfGE(x;1;)fGE(y;0+2;)dxdy=1 0+1+2;(13)p0G=P[Y1=Y2]=0 0+1+210fWE(z;0+1+2;)dz=0 0+1+2:(14)DiscriminationProcedureSupposef(X11;X21);


;(X1n;X2n)gisarandombivariatesampleofsizengeneratedeitherformaBVGE()orfromaMOBW().Basedontheabovesample,wewanttodecidefromwhichdistributionfunctionthedatasethasbeenobtained.Weusethefollowingnotationsandsetsfortherestofthepaper;I0=f(x1i;x2i);x1i=x2i=xi;i=1


;ng,I1=f(x1i;x2i);x1ix2i;i=1


;ng,I2=f(x1i;x2i);x1i&#x-278;&#x.066;x2i;i=1


;ng,I=I0[I1[I2,n0=I0,n1=I1andn2=I2,n0+n1+n2=n.Itisassumedthatn0=0,n1=0,andn2=0.Let=(0;1;2;)bethemaximumlikelihoodestimators(MLEs)of,basedontheassumptionthatthedatahavebeenobtainedfromBVGE().Similarly,let=(;0;1;2),betheMLEofbasedontheassumptionthatthedatahavebeenobtainedfromMOBW().Notethat(0;1;2;)and(;0;1;2)areobtainedbymaximizingthecorrespondinglog-likelihoodfunction,sayL1(0;1;2;)andL2(;0;1;2)respectively.Notethatherethelog-likelihoodoftheBVGEcanbewrittenas;L1()=(n0+2n1+2n2)ln+n1ln(0+1)+n1ln2(15)+(0+11)Xi21ln(1ex1)+(21)Xi21ln(1ex2)+n2ln1+n2ln(0+2)+(11)Xi22ln(1ex1)+(0+21)Xi22ln(1ex2)+n0ln0+(0+1+21)Xi20ln(1ex)(Xi20xi+Xi21[2x1i+Xi21[2x2i)(16)9 andthelog-likelihoodofBVWEcanbewrittenasL2()=(n0+2n1+2n2)ln+n1ln1+n2ln2+n0ln0+n1ln(0+2)+n2ln(0+1)+(1)Xi20lnx1i+Xi21[2lnx2i+Xi20lnxi1Xi21[2x1i+Xi20xi2Xi21[2x2i+Xi20xi0Xi22x1i+Xi21x2i+Xi20xi:(17)Weusethefollowingdiscriminationprocedure.Considerthefollowingstatistics:T=L2(;0;1;2)L1(0;1;2;):(18)IfT�0,wechooseMOBWdistribution,otherwisewepreferBVGEdistribution.Itmaybementionedthat(0;1;2;)and(;0;1;2)areobtainedbymaximizing(16)and(17)respectively.Computationallybotharequitechallengingproblems,andtomaximizedirectlyoneneedstosolveafourdimensionaloptimizationproblemineachcase.InboththecasesEMalgorithmcanbeusedquiteeectivelytocomputetheMLEsoftheunknownparameters.SeeforexampleKunduandGupta(2009)andKunduandDey(2009)forBVGEandMOBWdistributionsrespectively.Ineachcase,itinvolvessolvingjustaone-dimensionaloptimizationproblemateach`E'step,andboththemethodsworkquitewell.InthenextsectionweprovidetheasymptoticdistributionofT,whichwillhelptocomputetheasymptoticPCS.AsymptoticDistributionsInthissection,weusethefollowingnotations.Foranyfunctions,f1(U)andf2(U),EBVGE(f1(U)),VBVGE(f1(U))andCovBVGE(f2(U);f1(U))willdenotethemeanoff1(U),thevarianceoff1(U),andthecovarianceoff1(U)andf2(U)respectively,undertheassumptiontheUBVGE().Similarly,wedeneEBVWE(f1(U)),VBVWE(f1(U))andCovBVWE(f2(U);f1(U))10 asthemeanoff1(U),thevarianceoff1(U)andthecovarianceoff1(U)andf2(U)respec-tively,undertheassumptionthatUBVWE().Wehavethefollowingtwomainresults:Theorem1:UndertheassumptionthatdataarefromMOBW(;0;1;2)thedistribu-tionofTasdenedin(18),isapproximatelynormallydistributedwithmeanEMOBW(T)andvarianceVMOBW(T).TheexpressionsofEMOBW(T)andVMOBW(T)areprovidedbelow.ProofofTheorem1:ItisprovidedintheAppendix. NowweprovidetheexpressionsforEMOBW(T)andVMOBW(T).Wedenotelimn!1EMOBW(T) n=AMMOBW;andlimn!1EMOBW(T) n=AVMOBW:Therefore,limn!11 nEMOBW(T)=AMMOBW=EMOBW[ln(fMOBW(X1;X2;))ln(fBVGE(X1;X2;))]limn!11 nVMOBW(T)=AVMOBW=VMOBW[ln(fMOBW(X1;X2;))ln(fBVGE(X1;X2;))]:NotethatbothAMMOBWandAVMOBWcannotbeobtainedinexplicitform.Theyhavetobeobtainednumerically,andtheyarefunctionsofp1W,p2W,p3W,and.MoreoveritshouldbementionedthatthemisspeciedparameterasdenedinLemma1(Appendix)alsoneedstobecomputednumerically.Theorem2:UndertheassumptionthatdataarefromBVGE()thedistributionofTasdenedin(18),isapproximatelynormallydistributedwithmeanEBVGE(T)andvarianceVBVGE(T).TheexpressionsofEBVGE(T)andVBVGE(T)areprovidedbelow.ProofofTheorem2:ItisprovidedintheAppendix. NowweprovidetheexpressionsforEBVGE(T)andVBVGE(T).Inthiscasealsowedenotelimn!1EBVGE(T) n=AMBVGEandlimn!1VBVGE(T) n=AVBVGE:11 Therefore,limn!11 nEBVGE(T)=AMBVGE=EBVGE[ln(fMOBW(X1;X2;))ln(fBVGE(X1;X2;))]limn!11 nVBVGE(T)=AVBVGE=VBVGE[ln(fMOBW(X1;X2;))ln(fBVGE(X1;X2;))]:Asmentionedbefore,herealsobothAMBVGEandAVBVGEcannotbeobtainedinexplicitform.Theyhavetobeobtainednumerically,andtheyarealsofunctionsofp1G,p2G,p3G,and.ThemisspeciedparameterasdenedinLemma2(Appendix)alsoneedstobecomputednumerically.Basedontheasymptoticdistributions,itispossibletocomputetheprobabilityofcorrectselection(PCS)forboththecases.MisspecifiedParameterEstimatesEstimationofInthiscaseitisassumedthatthedatahavebeenobtainedfromMOBW()andwewouldliketocompute,themisspeciedBVGEparameters,asdenedinLemma1.Suppose(X1;X2)MOBW(),considerthefollowingevents:1=fX1X2g,2=fX1&#x-454;&#x.245;X2gand0=fX1=X2g.Moreover,1Aistheindicatorfunctiontakingvalue1atthesetAand0otherwise.Therefore,canbeobtainedastheargumentmaximumofEMOBW(ln(fBVGE(X1;X2;)))=1()(say),where1()=ln()+p1Wln(0+1)+p1Wln2+(0+11)EMOBW(ln(1eX1)1A1)+(21)EMOBW(ln(1eX2)1A1)EMOBW((X1+X2)1A1)+p2Wln1+(11)EMOBW(ln(1eX1)1A2)+p2Wln(0+2)+(0+21)EMOBW(ln(1eX2)1A2)EMOBW((X1+X2)IA2)+(0+1+21)EMOBW(ln(1eX)1A0)EMOBW(X1A0)+p0Wln0:12 Weneedtomaximize1()withrespecttoforxed,tocompute,numerically.Clearly,isafunctionof,butwedonotmakeitexplicitforbrevity.Sincemaximizing1()involvesafourdimensionaloptimizationprocess,wesuggesttouseanapproximateversionofit,whichcanbeperformedveryeasily,andworksquitewellinpractice.Theideabasicallycamefromthemissingvalueprinciple,andithasbeenusedbyKunduandGupta(2009)indevelopingtheEMalgorithm.Wesuggesttousethefollowing1(),the`pseudo'versionof1(),asfollows:1()=(p0W+u2p1W+w2p2W)ln0+(p0W+2p1W+2p2W)ln+(0+1+21)Eln(1eX1)1A0(E(X11A0)+E((X1+X2)1A1[A2))+(u1p1W+p2W)ln1+(w1p2W+p1W)ln2+(0+11)Eln(1eX1)1A1+(0+21)Eln(1eX2)1A2+(21)Eln(1eX2)1A1+(11)Eln(1eX1)1A2:Hereu1=0 0+2;u2=2 0+2;w1=0 0+1;w2=1 0+1;(19)p1W;p2W;p3Waresameasdenedbefore.TheexplicitexpressionsoftheexpectedvaluesareprovidedintheAppendix.Notethat1()isactually1()=limn!11 nEflpseudo(0;1;2;(X1i;X2i;i=1;


;n)g:(20)Herelpseudo()isthe`pseudo'log-likelihoodfunctionofthecompletedataset,asdescribedinKunduandGupta(2009).Moreover,ithasthesameformasinKunduandGupta(2009),butsincehereitisassumedthat(X1i;X2i)MOBW(;1;2;3),thereforetheexpressionsofu1;u2;w1;w2areas(19),andtheyaredierentthanKunduandGupta(2009).Nowthemaximizationof1()canbeperformedasfollows.Notethatforagiven,themaximizationof1()withrespectto0,1and2canoccurat0()=p0W+u2p1W+w2p2W E(ln(1eX1)1A0)+E(ln(1eX1)1A1)+E(ln(1eX2)1A2);13 1()=u1p1W+p2W E(ln(1eX1)1A0)+E(ln(1eX1)1A1)+E(ln(1eX1)1A2)2()=p1W+w1p2W E(ln(1eX2)1A0)+E(ln(1eX2)1A2)+E(ln(1eX2)1A1);respectively,andnallymaximizationof1()canbeobtainedbymaximizingprolefunc-tion,namely,1(0();1();2();)withrespecttoonly.Therefore,itinvolvessolvingaonedimensionaloptimizationproblemonly.EstimationofInthiscaseitisassumedthatthedatahavebeenobtainedfromBVGE()andwecompute,themisspeciedMOBWparameters,asdenedinLemma2.Inthiscase,canbeobtainedastheargumentmaximumofEBVGE(ln(fMOBW(X1;X2;)))=2()(say),where2()=(p0G+2p1G+2p2G)ln+p1Gln1+p2Gln2+p0Gln0+p1Gln(0+2)+p2Gln(0+1)+(1)[EBVGE(lnX11A1)+EBVGE(lnX11A2)]+(1)[EBVGE(lnX21A1)+EBVWE(lnX21A2)+EBVWE(lnX11A0)]1[EBVWE(X11A1)+EBVWE(X11A2)+EBVWE(X11A0)]2[EBVWE(X21A1)+EBVWE(X21A2)+EBVWE(X11A0)]0[EBVWE(X11A2)+EBVWE(X21A1)+EBVWE(X11A0)]Inthiscasealsoweneedtomaximize2()withrespecttonumericallytoobtain,foraxed.Clearly,dependson,andwedonotmakeitexplicitforbrevity.Similarly,asbeforesincemaximizationof2()involvesafourdimensionaloptimizationproblem,wesuggesttousethefollowingapproximationof2().Wesuggesttouse2()=(p0G+2p1G+2p2G)ln+(1)E(lnX11A0+(lnX1+lnX2)1A1[A2)0E(X11A0+X11A2+X21A1)+(p0G+a1p1G+1p2G)ln01E(X1)+(p1G+a2p2G)ln12E(X2)+(p2G+2p1G)ln1:14 Herea1=1 0+1;a2=0 0+2;b1=2 0+2;b2=0 0+2;p0G;p1G;p2Garesameasdenedbefore.TheexpressionsofthedierentexpectationsareprovidedintheAppendix.Itmaybesimilarlyobservedasbeforethat2()=limn!11 nE(lpseudo(;0;1;2(X1i;X2i);i=1;


;n));(21)where(X1i;X2i)BVGE(0;1;2;).Theexplicitexpressionoflpseudo()isavailableinKunduandDey(2009).Themaximizationof2()withrespecttocanbeperformedquiteeasily.Forxedthemaximization2()withrespectto1,2and0,canbeobtainedfor1=p1G+2p2G E(X1)2=p2G+a2p1G E(X2)0=p0G+a1p1G+1p2G E(X11A0)+E(X11A2)+E(X21A1);respectively,andnallythemaximization2()canbeperformedbymaximizingtheprolefunction2(;0();1();2())withrespecttoonly.NumericalResultsInthissectionweperformsomenumericalexperimentstoobservehowtheseasymptoticre-sultsworkfordierentsamplesizes,andfordierentparametervalues.Allthesecomputa-tionsareperformedattheIndianInstituteofTechnologyKanpur,usingIntel(R)Core(TM)2QuadCPUQ95502.83GHz,3.23GBRAMmachines.TheprogramsarewritteninRsoft-ware(2.8.1).Theycanbeobtainedfromtheauthorsonrequest.WecomputePCSbased15 onMonteCarlosimulation(MC),andalsobasedontheasymptoticresults.Wereplicatetheprocess1000,timesandcomputetheproportionofcorrectselection.ForcomputingthePCSbasedonasymptoticresults,rstwecomputethemisspeciedparametersandbasedonthosemisspeciedparameterswecomputethePCS.Case1:ParentDistributionisMOBWInthiscaseweconsiderthefollowingparametersets;Set1:=2.0,0=1.0,1=1.0,2=1.0;Set2:=1.5,0=1.0,1=1.0,2=1.0;Set3:=1.5,0=0.5,1=0.5,2=0.5;Set4:=1.5,0=2.0,1=1.0,2=1.5,anddierentsamplesizesnamelyn=20,40,60,80,100.Foreachparametersetandforeachsamplesize,wehavegeneratedthesamplefromMOBWdistribution.ThenwecomputetheMLEsoftheunknownparametersandtheassociatedlog-likelihoodvalues,assumingthatthedataarecomingfromMOBWorBVGEdistribution.IncomputingtheMLEsoftheunknownparameters,wehaveusedtheEMalgorithmassuggestedinKunduandDey(2009)andKunduandGupta(2009)respectively.Finallybasedonthemaximizedlog-likelihoodvalueswedecidewhetherwehavemadethecorrectdecisionornot.Wereplicatetheprocess1000times,andcomputetheproportionofcorrectselection.TheresultsarereportedintherstrowsoftheTables1to4.Table1:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet1. n 20 40 60 80 100 MC 0.9255 0.9808 0.9953 0.9987 0.9997 AD 0.9346 0.9837 0.9956 0.9987 0.9996 Nowtocomparetheseresultswiththecorrespondingasymptoticresults,rstwecomputethemisspeciedparametersforeachparameterset,andtheyarepresentedinthefollowing16 Table2:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet2. n 20 40 60 80 100 MC 0.9255 0.9808 0.9953 0.9987 0.9997 AD 0.9212 0.9772 0.9928 0.9976 0.9992 Table3:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet3. n 20 40 60 80 100 MC 0.9073 0.9749 0.9914 0.9979 0.9989 AS 0.9204 0.9767 0.9926 0.9975 0.9992 Table5.IneachcaseweneedtocomputeAMMOBWandAVMOBW,asdenedinTheorem1.SincetheexactexpressionsofAMMOBWandAVMOBWarequitecomplicated,wehaveusedsimulationconsistentestimatesofAMMOBWandAVMOBW,whichcanbeobtainedveryeasily.ThesimulationconsistentestimatorsofAMMOBWandAVMOBWareobtainedusing10,000replications,andtheyarereportedinTable6NowusingTheorem1,basedontheasymptoticdistributionofT,thediscriminationstatistic,wecomputetheprobabilityofcorrectselection,i:e:P(T�0)fordierentsamplesizes.TheresultsarereportedinthesecondrowsofTables1to4foralltheparametersets.Itisveryinterestingtoobservethatforthebivariatecase,evenforsmallsamplesizestheprobabilityofcorrectselectionsareveryhigh,andtheasymptoticresultsmatchverywellwiththesimulatedresults.Case2:ParentDistributionisBVGEInthiscaseweconsiderthefollowingparametersets;Set5:0=1.5,1=2.0,2=1.0,=1.0;Set6:0=1.0,1=1.0,2=1.0,=1.0;17 Table4:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet4. n 20 40 60 80 100 MC 0.8834 0.9587 0.9843 0.9952 0.9973 AS 0.8996 0.9648 0.9866 0.9947 0.9979 Table5:Misspeciedparametervaluesfordierentparametersets. Set 1 2 0  1 1.5098 1.5098 1.6228 2.81 2 0.8853 0.8853 0.9458 2.35 3 0.8908 0.8908 0.9600 1.49 4 1.3393 1.0782 0.7362 3.10 Set7:0=2.0,1=2.0,2=2.0,=1.0;Set8:0=1.5,1=1.5,2=1.5,=1.0,andthesamesamplesizesasinCase1.InthiscasewegeneratethesamplefromBVGE,andusingthesameprocedureasbeforewecomputetheproportionofcorrectselection.TheresultsarereportedintherstrowsoftheTables7to10.Nowtocomputetheasymptoticprobabilityofcorrectselection,rstwecomputethemisspeciedparametersassuggestedinsection5,andtheyarereportedinTable11.WealsoreportsimulatedconsistentestimatesofAMBVGEandAVBVGEinTable12.Nowsimilarlyasbefore,basedontheasymptoticdistributionofT,asprovidedinTheo-rem2,wecomputetheprobabilityofcorrectselectioninthiscase,i:e:P(T0)fordierentsamplesizes.WereporttheresultsinthesecondrowsofTables7to10foralltheparametersets.Inthiscasealso,itobservedthattheasymptoticresultsmatchextremelywellwiththesimulatedresults.18 Table6:AMMOBWandAVMOBWfordierentparametersets. Set AMMOBW AVMOBW 1 0.2346 0.4823 2 0.1982 0.3936 3 0.2297 0.4317 4 0.1762 0.4317 Table7:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet5. n 20 40 60 80 100 MC 0.9195 0.9797 0.9935 0.9986 0.9993 AS 0.9330 0.9830 0.9953 0.9986 0.9996 7DataAnalysis:Inthissectionwepresenttheanalysisofarealdatasetforillustrativepurposes.ThesedataarefromtheNationalFootballLeague(NFL),AmericanFootball,matchesplayedonthreeconsecutiveweekendsin1986.Ithasbeenoriginallypublishedin`WashingtonPost'.Inthisbivariatedataset,thevariablesarethe`gametime'totherstpointsscoredbykickingtheballbetweengoalposts(X1)andthe`gametime'totherstpointsscoredbymovingtheballintotheendzone(X2).Thesetimesareofinteresttoacasualspectatorwhowantstoknowhowlongonehastowaittowatchatouchdownortoaspectatorwhoisinterestedonlyatthebeginningstagesofagame.Thedata(scoringtimesinminutesandseconds)arerepresentedinTable13.Wehaveanalyzedthedatabyconvertingthesecondstothedecimalminutes,i:e:2:03hasbeenconvertedto2.05.ThevariablesX1andX2havethefollowingstructure:(i)X1X2meansthattherstscoreisaeldgoal,(ii)X1=X2meanstherstscoreisaconvertedtouchdown,(iii)19 Table8:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet6. n 20 40 60 80 100 MC 0.9001 0.9701 0.9892 0.9962 0.9984 AS 0.9153 0.9741 0.9914 0.9970 0.9989 Table9:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet7. n 20 40 60 80 100 MC 0.9189 0.9811 0.9944 0.9987 0.9994 AS 0.9347 0.9837 0.9955 0.9987 0.9996 X1�X2meanstherstscoreisanunconvertedtouchdownorsafety.Inthiscasethetiesareexactbecauseno`gametime'elapsesbetweenatouchdownandapoint-afterconversionattempt.Therefore,itisclearthatinthiscaseX1=X2occurswithpositiveprobability,andsomesingulardistributionshouldbeusedtoanalyzethisdataset.Ifwedenethefollowingrandomvariables:U1=timetorsteldgoalU2=timetorstsafetyorunconvertedtouchdownU0=timetorstconvertedtouchdown,then,X1=minfU0,U1gandX2=minfU0,U2g.Therefore,(X1;X2)hasasimilarstructureastheMarshall-Olkinbivariateexponentialmodel.CsorgoandWelsh(1989)analyzedthedatausingtheMarshall-Olkinbivariateexponentialmodelbutconcludedthatitdoesnotworkwell,becauseX2maybeexponentialbutX1isnot.InfactitisobservedthattheempiricalhazardfunctionsofbothX1andX2areincreasingfunctions.SincebothMOBWandBVGEcanhaveincreasingmarginalhazardfunctions,wet20 Table10:ProbabilityofcorrectselectionbasedonMonteCarlo(MC)simulationsandbasedonasymptoticdistribution(AD)forparameterSet8. n 20 40 60 80 100 MC 0.9096 0.9768 0.9929 0.9975 0.9991 AS 0.9299 0.9816 0.9947 0.9984 0.9995 Table11:Misspeciedparametervaluesfordierentparametersets. Set  0 1 2 5 1.6199 0.1732 0.1137 0.1992 6 1.4199 0.2575 0.2418 0.2418 3 1.8200 0.1123 0.1050 0.1050 8 1.6199 0.1665 0.1553 0.1553 boththemodelstothedataset.ForMOBWdistributionusingEMalgorithmassuggestedinKunduandDey(2009),wecomputetheMLEsoftheunknownparametersas=1.2889,0=11.2073,1=8.3572,2=0.4720,andtheassociated95%condenceintervalsare(1.0372,1.5406),(5.7213,16.6932),(2.5312,14.1831),(-0.4872,1.4314)respectively.Thecorrespondinglog-likelihoodvalueis47.8041.IncaseofBVGEdistributionusingtheEMalgorithmassuggestedinKunduandGupta(2009),weobtainedtheMLEsoftheunknownparametersas0=1.1628,1=0.0558,2=0.5961,=9.5634,andtheassociated95%condenceintervalsare(0.6991,1.6266),(-0.0205,0.1322),(0.2751,0.9171)and(6.5298,12.5970)respectively.Thecorrespondinglog-likelihoodvalueis38.0042.Therefore,basedonthelog-likelihoodvaluesweprefertousetheMOBWmodelratherthanBVGEmodeltoanalyzethisdataset.Nowtocomputetheprobabilityofcorrectselectioninthiscase,weperformnon-parametricbootstrap.Thehistogramofthebootstrapsampleofthediscriminationstatisticsispro-videdinFigure3.Basedononethousandbootstrapreplications,itisobservedthatthe21 Table12:AMMOBWandAVMOBWfordierentparametersets. Set AMMOBW AVMOBW 5 0.2224 0.4406 6 0.1967 0.4095 3 0.2316 0.4692 8 0.2128 0.4157 X1 X2 X1 X2 X1 X2 2:03 3:59 5:47 25:59 10:24 14:15 9:03 9:03 13:48 49:45 2:59 2:59 0:51 0:51 7:15 7:15 3:53 6:26 3:26 3:26 4:15 4:15 0:45 0:45 7:47 7:47 1:39 1:39 11:38 17:22 10:34 14:17 6:25 15:05 1:23 1:23 7:03 7:03 4:13 9:29 10:21 10:21 2:35 2:35 15:32 15:32 12:08 12:08 7:14 9:41 2:54 2:54 14:35 14:35 6:51 34:35 7:01 7:01 11:49 11:49 32:27 42:21 6:25 6:25 5:31 11:16 8:32 14:34 8:59 8:59 19:39 10:42 31:08 49:53 10:09 10:09 17:50 17:50 14:35 20:34 8:52 8:52 10:51 38:04 Table13:AmericanFootballLeague(NFL)dataprobabilityofcorrectselectionis0.98.ConclusionInthispaperwehaveconsidereddiscriminationbetweentwosingularbivariatedistributionsnamely,MOBWandBVGEdistributions.Boththedistributionshavesingularpartandabsolutecontinuouspart.Thedierenceofthemaximizedlog-likelihoodvalueshasbeenusedasthediscriminationstatistic.Wehaveobtainedtheasymptoticdistributionofthe22 -10-5 0 5 10 15 20 Figure3:Histogramofthebootstrapsampleofthediscriminationstatistic.discriminationstatistic,whichcanbeusedtocomputetheasymptoticprobabilityofcor-rectselection.MonteCarlosimulationsareperformedtoseethebehavioroftheproposedmethod.ItisknownthatthediscriminationbetweenWeibullandgeneralizedexponentialdistributionsisquitedicult,seeGuptaandKundu(2003),butinthispaperitisobservedthatthediscriminationbetweenMOBWandBVGEisrelativelymucheasier.Evenwithsmallsamplesizestheprobabilityofcorrectselectionisquitehigh.Moreovertheasymptoticprobabilityofcorrectselectionmatchesverywellwiththesimulatedprobabilityofcorrectselectionevenformoderatesamplesizes.Wehaveperformedtheanalysisofadataset,andcomputedtheprobabilityofcorrectselectionusingnon-parametricbootstrapmethod.Althoughwedonothaveanytheoreticalresults,itseemsnon-parametricbootstrapmethodalsocanbeusedquiteeectivelyincomputingtheprobabilityofcorrectselectioninthiscase.Moreworkisneededinthisdirection.Appendix:ToproveTheorem1,weneedthefollowingLemma1.Herea:s:!meansconvergesalmostsurely.Lemma1:UndertheassumptionthatdataarefromBVWE(;0;1;2),asn!1,we23 have(i)a:s:!,0a:s!0,1a:s!1and2a:s!2wherefor=(;0;1;2)EMOBW(ln(fMOBW(X1;X2;)))=maxEMOBW(ln(fMOBW(X1;X2;)))(22)(ii)0a:s!0,1a:s!1,2a:s!2,a:s!,wherefor=(0;1;2;),EMOBW(ln(fBVGE(X1;X2;)))=maxEMOBW(ln(fBVGE(X1;X2;)))(23)Itmaybenotedthatmaydependon,butwedonotmakeitexplicitforbrevity.(iii)IfwedenoteT=L2(;0;1;2)L1(0;1;2;)thenn1 2[TEMOBW(T)]isasymptoticallyequivalentton1 2[TEMOBW(T)]ProofofLemma1:ItisquitestandardanditfollowsalongthesamelineastheproofofLemma2.2ofWhite(1982),anditisavoided. ProofofTheorem1:UsingCentrallimittheoremandpart(ii)ofLemma1,itfollowsthatn1 2[TEBVWE(T)]isasymptoticallynormallydistributedwithmeanzeroandvari-anceVMOBW(T).Therefore,usingpart(iii)ofLemma1,theresultimmediatelyfollows. ToproveTheorem2,andfordeningthemisspeciedparameterweneedthefollowingLemma2,whoseproofissameastheproofofLemma1. Lemma2:SupposethedatafollowBVGE(0;1;2;),asn!1,wehave(i)0a:s:!0,1a:s:!1,^2a:s:!2and^!whereEBVGE(ln(fBVGE(X1;X2;)))=maxEBVGE(ln(fBVGE(X1;X2;)))(24)24 (ii)a:s!,0a:s!0,1a:s!1,2a:s!2,where=(;0;1;2)EBVGE(ln(fMOBW(X1;X2;)))=maxEBVGE(ln(fMOBW(X1;X2;)))(25)Herealsodependon,butwedonotmakeitexplicitforbrevity.(iii)IfwedenoteT=L2(;0;1;2)L1(0;1;2;)thenn1 2[TEBVGE(T)]isasymptoticallyequivalentton1 2[TEBVGE(T)]ProofofTheorem2:AlongthesamelineastheProofofLemma1,italsofollowsusingLemma2. ThefollowingLemmaswillbeusefulincomputingthedierentexpectedvaluesneededin1()andin2().Here1A0,1A1and1A2aresameasdenedbefore.LemmaA.1:LetW0GE(0+1+2;),W1GE(0+1;),W2GE(0+2;)and(X1;X2)BVGE(0;1;2;).Ifg()isanyBorelmeasurablefunction,thenE(g(X1)1A1)=E(g(W1))+0+1 0+1+2E(g(W0)):E(g(X1)1A2)=1 0+1+2E(g(W0)):E(g(X1)1A0)=E(g(X2)1A0)=0 0+1+2E(g(W0)):E(g(X2)1A1)=2 0+1+2E(g(W0)):E(g(X2)1A2)=E(g(W2))+0+2 0+1+2E(g(W0)):ProofofLemmaA.1:SeeKunduandGupta(2009). LemmaA.2:LetZ0WE(,0+1+2),Z1WE(,0+1),Z2WE(,0+2)and(X1;X2)MOBW(;0;1;2).Ifg()isanyBorelmeasurablefunction,thenE(g(X1)1A1)=1 0+1+2E(g(Z1)):25 E(g(X1)1A2)=E(g(Z1))0+1 0+1+2E(g(Z0)):E(g(X1)1A0)=E(g(X2)1A0)=0 0+1+2E(g(Z0)):E(g(X2)1A1)=E(g(Z2))0+2 0+1+2:E(g(X2)1A2)=2 0+1+2E(g(Z2)):ProofofLemmaA.1:TheycanbeobtainedalongthesamelineasinLemmaA.1. 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