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First,IoermysincerestgratitudetomycommitteechairDr.MalayGhosh,whosupportedmewithhisknowledgeandpatience.IwouldliketothankmysupervisorycommitteemembersDr.RamonLittell,Dr.BhramarMukherjeewhoalsoservedascochairandDr.JonathanShuster.SpecialthanksgotoDr.BhramarMukherjeeforhercontinuousguidance,supportandhelp.IacknowledgeherandDr.DalhoKimfordoingthesimulationstudiesinmydissertation.FinallyIwouldliketothankmyfamily,especiallymyhusbandSwadeshmukulSantra,whorstencouragedmetopursuethisdegree,andmydaughterLaboniSantra.Withouttheircontinuingsupportandencouragement,Iwouldnothavenishedthisdegree. 4
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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1LITERATUREREVIEW .............................. 11 1.1Introduction ................................... 11 1.2MatchingViaPosteriorQuantiles ....................... 15 1.2.1NotationandDierentialEquation .................. 15 1.2.2SpecialCases .............................. 16 1.2.2.1Casep=1 ........................... 16 1.2.2.2Casep=2 ........................... 18 1.2.3OrthogonalParameterization ...................... 19 1.2.4Examples ................................. 20 1.3MatchingPriorsforDistributionFunctions .................. 23 1.3.1NotationandDierentialEquation .................. 23 1.3.2OrthogonalParameterization ...................... 24 1.3.3Examples ................................. 25 1.4MatchingPriorsforHighestPosteriorDensityRegions ........... 25 1.4.1NotationandDierentialEquation .................. 26 1.4.2SpecialCase:p=1 ............................ 27 1.4.3OrthogonalParameterization ...................... 28 1.4.4Examples ................................. 28 1.5MatchingPriorsAssociatedwithOtherCredibleRegions .......... 29 1.5.1MatchingPriorsAssociatedwiththeLRStatistic .......... 30 1.5.1.1Introduction .......................... 30 1.5.1.2Dierentialequation ..................... 30 1.5.1.3Specialcase:p=1 ....................... 30 1.5.1.4Nuisanceparametersandorthogonality ........... 31 1.5.2MatchingPriorsAssociatedwithRao'sScoreandWald'sStatistic 32 1.5.2.1Introduction .......................... 32 1.5.2.2Dierentialequation ..................... 33 1.5.2.3Specialcase:p=1 ....................... 34 2MATCHINGPRIORSFORSOMEBIVARIATENORMALPARAMETERS .. 35 2.1TheOrthogonalReparameterization ..................... 35 2.2QuantileMatchingPriors ............................ 41 5
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..................... 43 2.4HighestPosteriorDensity(HPD)MatchingPriors .............. 44 2.5MatchingPriorsViaInversionofTestStatistics ............... 46 2.6ProprietyofPosteriorsandSimulationStudy ................. 47 3THEBIVARIATENORMALCORRELATIONCOEFFICIENT ......... 52 3.1TheOrthogonalParameterization ....................... 52 3.2QuantileMatchingPriors ............................ 56 3.3HighestPosteriorDensity(HPD)MatchingPriors .............. 57 3.4MatchingPriorsViaInversionofTestStatistics ............... 58 3.5ProprietyofthePosteriors ........................... 59 3.6LikelihoodBasedInference ........................... 61 3.7SimulationStudy ................................ 66 4RATIOOFVARIANCES .............................. 74 4.1TheOrthogonalParameterization ....................... 74 4.2QuantileMatchingPriors ............................ 77 4.3MatchingViaDistributionFunctions ..................... 78 4.4HighestPosteriorDensity(HPD)MatchingPriors .............. 79 4.5MatchingPriorsViaInversionofTestStatistics ............... 80 4.6ProprietyofthePosteriors ........................... 80 4.7SimulationStudy ................................ 83 5SUMMARY ...................................... 91 REFERENCES ....................................... 93 BIOGRAPHICALSKETCH ................................ 98 6
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Table page 1-1Fisher-VonMisesP(;)(0:05;) ........................... 34 1-2Fisher-VonMisesP(;)(0:95;) ........................... 34 2-1FrequentistCoverageProbabilitiesof95%HPDIntervalsfor,andwhen21=1and22=1 .................................. 51 3-1SimulationComparingPriorsforBivariateNormalCorrelationCoecient ... 69 4-1SimulationComparingPriorsSuggestedforBivariateNormalRatioofStandardDeviation ....................................... 85 7
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Figure page 3-1PlotofGelman-RubinDiagnosticStatisticforUnderPriorIIIforn=10UndertheSimulationSettingofSection3.7. ........................ 68 3-2SampleTracePlotforAlltheParametersunderPriorIIIforn=10UndertheSimulationSettingofSection3.7 .......................... 70 3-3PosteriorDistributionforunderPriorIforDierentSampleSizes,UndertheSimulationSettingofSection3.7 .......................... 71 3-4PosteriorDistributionforunderPriorIIforDierentSampleSizes,UndertheSimulationSettingofSection3.7 .......................... 72 3-5SamplePosteriorDistributionforunderPriorIIIforDierentSampleSizes,UndertheSimulationSettingofSection3.7 .................... 73 4-1SampleTracePlotforallParametersunderPrior3undertheSimulationSettingofSection4.8 ..................................... 86 4-2PlotofGelman-RubinDiagnosticStatisticfor1underPrior3 .......... 87 4-3PosteriorDistributionfor1underPrior1forDierentSampleSizes ...... 88 4-4PosteriorDistributionfor1underPrior2forDierentSampleSizes ...... 89 4-5SamplePosteriorDistributionfor1underPrior3forDierentSampleSizes .. 90 8
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Inpractice,mostBayesiananalysesareperformedwithsocalled\non-informative"priors.Thisisespeciallysowhenthereislittleornopriorinformation,andyettheBayesiantechniquecanleadtosolutionssatisfactoryfromboththeBayesianandthefrequentistperspectives.Thestudyofprobabilitymatchingpriorsensuring,uptothedesiredorderofasymptotics,theapproximatefrequentistvalidityofposteriorcrediblesetshasreceivedsignicantattentioninrecentyears.Inthisdissertationwedevelopsomeobjectivepriorsforcertainparametersofthebivariatenormaldistribution.Theparametersconsideredaretheregressioncoecient,thegeneralizedvariance,theratioofoneoftheconditionalvariancestothemarginalvarianceoftheothervariable,thecorrelationcoecientandtheratioofthestandarddeviations.ThecriterionusedistheasymptoticmatchingofcoverageprobabilitiesofBayesiancredibleintervalswiththecorrespondingfrequentistcoverageprobabilities.Variousmatchingcriteria,namely,quantilematching,matchingofdistributionfunctions,highestposteriordensitymatching,andmatchingviainversionofteststatisticsareused. Oneparticularpriorisfoundwhichmeetsallthematchingcriteriaindividuallyfortheregressioncoecient,thegeneralizedvarianceandtheratioofoneoftheconditionalvariancestothemarginalvarianceoftheothervariable.Forthecorrelationcoecientthough,eachmatchingcriterionleadstoadierentprior.Therehowever,doesnotexista 9
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Theproprietyoftheresultantposteriorsisprovedineachcaseundermildconditionsandsimulationresultssuggestthattheapproximationsarevalidevenformoderatesamplesizes.Further,severallikelihoodbasedmethodshavebeenconsideredforthecorrelationcoecient.Onecommonfeatureofallthesemodiedlikelihoodsisthattheyarealldependentonthedataonlythroughthesamplecorrelationcoecientr. 10
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TheearliestuseofnoninformativepriorsisattributedtoLaplace(1812).Laplace'srule,ortheprincipleofinsucientreason,assignsaatpriorovertheentireparameterspace.Aproblemwiththisruleisthatitisnotinvariantunderone-toonereparameterization.Forexample,ifisgivenauniformdistribution,then=exp()willnothaveauniformdistribution.Conversely,ifwestartwithauniformdistributionfor,then=log()willnothaveauniformdistribution.Sincemoststatisticalmodelsdonothaveauniqueparameterization,thisbecomesbothersome.Forexample,auniformpriorforthestandarddeviationwillnottransformintoauniformpriorforthevariance2.Thislackofinvarianceoftheuniformprioroftentranslatesintosignicantvariationintheresultingposteriors. Thus,Jereys(1961)proposedapriorwhichremainsinvariantunderanyone-to-onereparameterization.Inthegeneralmultiparametersetup,writingtheFisherInformationmatrixasI(),whereI()=E@2l @i@j 2:
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2; whereJ=det(@ @).Sothepriorsdenedbytheruleonandtransformaccordingtothechange-of-variablesformula.Thusitdoesnotrequiretheselectionofanyspecicparameterization. TherearemanyintuitivejusticationstouseJerey'sprior.Onethatconcernsusisaprobabilitymatchingproperty.Asanexample,ifX1;:::;XnareiidN(;1),thenXn=Pni=1Xi=nistheMLEof.Withtheuniformprior()/c(aconstant),theposteriorofisN(Xn;1=n).Accordingly,writingzfortheupper100pointoftheN(0,1)distribution,P(Xn+zn1=2jXn)=1=P(Xn+zn1=2j) Thisisanexampleofexactmatching.OtherexamplesofexactmatchingcanbefoundinDatta,Ghosh,MandMukerjee(2000)andSeverini,MukerjeeandGhosh,M.(2002).However,inmostinstancesonehastorelyonasymptoticsratherthanexactmatching.Toseethis,suppose^nistheMLEof.Then^njisasymptoticallyN(;I1()),whereI()istheFisherInformationnumber.Usingthetransformationg()=RI1=2(t),g(^n)isasymptoticallyN(g();1)bythedeltamethod.Now,intuitivelyoneexpectstheuniformpriorastheasymptoticmatchingpriorforg().Transformingbacktotheoriginalparameter,Jereys'priorisaprobabilitymatchingpriorfor.ThisisdiscussedinGhosh,M.(2001).Toseethis,let=g()and()=1.Thenj@ @j=jg0()j=I1=2(). Theabovematchingpropertyisusuallyreferredtoasthequantilematchingproperty.However,quantilematchingisoneofseveralmatchingcriteriaavailableintheliterature.Typically,thismatchingofposteriorcoverageprobabilityofaBayesiancrediblesetwiththecorrespondingfrequentistcoverageprobabilityisalsoaccomplishedthrougheither(a) 12
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MatchingpriorsbasedonposteriorquantileswasrstinvestigatedbyWelchandPeers(1963)whoconsideredascalarparameterofinterestintheabsenceofanynuisanceparameters.Inthiscasetheyshowedbysolvingadierentialequationthatthefrequentistcoverageprobabilityofaone-sidedposteriorcredibleintervalformatchesthenominallevelwitharemainderofo(n1 2),wherenisthesamplesize,ifandonlyifoneworkswithJereys'prior.Suchapriorwillbereferredtoasarstorderprobabilitymatchingprior.WelchandPeersprovedthisresultonlyforcontinousdistributions.Ghosh,J.K.(1994)pointedoutasuitablemodicationwhichwouldleadtothesameconclusionfordiscretedistributions.Ontheotherhand,ifonerequirestheremaindertobeoftheordero(n1),thenwehaveasecondorderprobabilitymatchingprior.WeshallseelaterthattheJereys'priorisnotnecessarilyasecondordermatchingpriorevenintheoneparametercase.Moreover,Jerey'spriorhasbeencriticizedinthepresenceofnuisanceparameters.Forexample,Bernardo(1979)hasshownthatJerey'spriorcanleadtomarginalizationparadox(cf.Dawid,StoneandZidek(1973))forinferenceabout whenthemodelisnormalwithmeanandvariance2.Asecondexample,duetoBergerandBernardo(1992a),showsthatJerey'spriorcanleadtoaninconsistentestimatoroftheerrorvarianceinthebalancedone-waynormalANOVAmodelwhenthenumberofcellsgrowstoinnityindirectproportiontothesamplesize.SoJerey'spriorfailstoavoidtheNeyman-Scott(1948)phenomenon. TheoriginalideaofWelchandPeers(1963)waspursuedinthenuisanceparametercasebyPeers(1965),Stein(1985),Tibshirani(1989),Nicolaou(1993),DattaandGhosh,J.K.(1995a,b),DattaandGhosh,M.(1995,1996),Ghosh,M.,CarlinandSrivastava(1995),Ghosh,M.andYang(1996),Datta(1996)andGarvanandGhosh,M.(1996)amongothers.Asweshallsee,matchingisobtainedbysolvingdierentialequations.Thecalculationsarehighlysimpliediftheparameterofinterestisorthogonaltothenuisance 13
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@s@l @t=1 @s@t=0 fors=1;:::;p1;t=p1+1;:::;p1+p2;thisistoholdforallintheparameterspace.Notethatl()isthelog-likelihoodandireferstoinformationperobservation,whichwillbeassumedtobeO(1)asn!1. Supposethatwehaveascalarparameterofinterest,thatis,=(1;:::;p)T,where1istheparameterofinterestandtherestarenuisanceparameters.WritingI()=((Ijk))astheFisherInformationmatrix,if1isorthogonalto(2;:::;pT),thatisI1k=0forallk=2;:::;p,extendingthepreviousintuitiveargument,()/I111=2()isaprobabilitymatchingprior.Inthepresenceofnuisanceparameters,Jereys'priormaynotsatisfythequantilematchingproperty.Asanexample,considertheBehrens-Fisherproblem(Ghosh,M.andKim,2001).Themodelisrepresentedbythedensity1 (1=12)+(1=22);3=1;4=2 14
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Inthefollowingsectionswereviewandcharacterizethedierentmatchingpriors. holdsforr=1or2andforeach(0<<1).Herenisthesamplesize,=(1;:::;p)Tisanunknownparametervector,1istheone-dimensionalparameterofinterest,Pf:gisthefrequentistprobabilitymeasureunder,and1(1)(;X)isthe(1)thposteriorquantileof1,under(:),giventhedataX.Priorssatisfying( 1:2:1 )forr=1or2arecalledrstorsecondordermatchingpriorsrespectively.Clearly,theyensurethatone-sidedBayesiancrediblesetsoftheform(;1(1)(;X)]for1havecorrectfrequentistcoverageaswelluptotheorderofapproximationindicatedin( 1:2:1 ).Inthepresenceofnuisanceparameters,arstordermatchingpriorisnotunique.Thestudyofsecondordermatchingpriors,whichensurescorrectfrequentistcoveragetoahigherorderofapproximation,canhelpsignicantlyinnarrowingdowntheclassofcompetingrstordermatchingpriors. 15
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LetI=((Ijr))betheperobservationFisherinformationmatrixat.DeneI1=((Ijr)),then Theseareconsideredtobesmoothfunctionsof.Also,for1j;rp,let Nowwegivethetheoremwhichcharacterizestherstandsecondorderprobabilitymatchingpriors. 2Ij1()g=0(1.2.1.4) (b)Aprior(:)issecondorderprobabilitymatchingifandonlyifitsatises,inaddition,thepartialdierentialequation 3[DufjrLjrs(3su+su)g]pXj=1pXr=1DjDrfjrg=0(1.2.1.5) Part(a)wasprovedoriginallybyPeers(1965)andpart(b)byMukerjeeandGhosh,M.(1997). 1:2:1:4 ) 16
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df()=I1=2g=0; whichistheJereys'(1961)prior.Thusinthiscase,Jereys'prioristheuniquerstordermatchingprior.Furthermore,forp=1,by( 1:2:1:1 )and( 1:2:1:2 ),( 1:2:1:4 )reducesto 1 3f()L111=I2gd df()=Ig=constant:(1.2.2.2) NowforJereys'prior,givenby( 1:2:2:1 ),andusingthestandardregularitycondition,itfollowsfromBartlett(1953)that dI=(L1;11+L111);(1.2.2.3) and Thus,thelefthandsideof( 1:2:2:2 )simpliesto 1 3I3=2L111d dI1=2=1 6I3=2L1;1;1(1.2.2.5) Summarizingtheaboveresultswegetthefollowingtheorem. (b)Furthermore,itisalsosecondorderprobabilitymatchingifandonlyifI3=2L1;1;1isaconstantfreefrom. Apartfromthisearlyresultonmatchingpriors,anotherresult,againduetoWelchandPeers(1963),ispresentedinthenexttheorem. 17
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Intheoneparameterscalemodel );(1.2.2.7) where>0andf(:)isadensitywithsupporteitherR1or[0;1),Jereys'priorgivenby()/1issecondorderprobabilitymatching.Inthiscasetoo,itcanbeshownthatthematchingisexact.Evenbeyondthestandardoneparameterlocationorscalemodels,Jereys'priorcanenjoythesecondordermatchingproperty.OntheotherhandtherecanbemodelswheretheconditioninTheorem1.2.2(b)doesnotholdandconsequentlynosecondordermatchingpriorisavailable. (12)3: 1:2:1:2 ),here where 18
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1:2:2:8 ),thepartialdierentialequations( 1:2:1:4 )and( 1:2:1:5 ),forrstandsecondorderprobabilitymatching,canbeexpressedas and 1 3D1f()Q2(L1113L112+3L1222L2223)g1 3D2f()Q2(L1113L112+3L1222L2223)gD2f()QI122(L1122L122+L2222)gD12f()Qg+2D1D2f()QgD22f()Q2g=0;(1.2.2.11) respectively.Thesecondordermatchingcondition( 1:2:2:11 )forthecasep=2isduetoMukerjeeandDey(1993). identicallyin.ThenI1j=0;2jp,andby( 1:2:1:2 ),11=I11=I111;jr=0;if(j;r)6=(1;1);jr=0;ifeitherj=1orr=1;jr=Ijr;ifj2andr2: 1:2:1:4 )and( 1:2:1:5 ),forrstandsecondorderprobabilitymatching,canbeexpressedas 19
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3D1f()I211L111gD21f()I111g=0;(1.2.3.3) respectively. Aprior(:)satises( 1:2:3:2 )andishencerstorderprobabilitymatchingifandonlyifitisoftheform whered(.)(>0)isanysmoothfunctionof(2)=(2;:::;p)T.ThisresultisduetoTibshirani(1989).Nicolaou(1993)alsoproveditusinganotherapproach.By( 1:2:3:3 ),aprioroftheform( 1:2:3:4 )issecondorderprobabilitymatchingifandonlyif 6d((2))D1fI3=211L1;1;1g=0:(1.2.3.5) wherew=R10(ulogu)exp(u)duand1;2>0.Then1istheparameterofinterestand2isorthogonalto1.Also,I11/12;I22=12=22;L1;1;1/13;L112/(12)1: 1:2:3:4 ),therefore,rstordermatchingisachievedifandonlyif()=d(2)=1.Moreover,by( 1:2:3:5 )suchapriorissecondordermatchingifandonlyifd(2)/21. 20
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where1;2>0.Then1istheparameterofinterestandonecancheckthat( 1:2:4:2 )isanorthogonalparameterization.Furthermore, 1:2:3:4 ),rstordermatchingisachievedifandonlyif()=d(2)2 1:2:3:5 )suchapriorissecondordermatchingifandonlyif,inaddition,d(2)isaconstant.Thus()/2=(12+1)istheuniquesecondordermatchingprior.Interestinglyundertheoriginalparameterization,thisgetstransformedtotheuniformprioron(1;2). 21
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1:2:4:3 )isanorthogonalparameterization.Furthermore, 22(412+22)3=2:(1.2.4.4) By( 1:2:3:4 ),rstordermatchingisachievedifandonlyif Suchapriorisalsosecondordermatchingifandonlyifd(2)satises( 1:2:3:5 )which,inviewof( 1:2:4:4 ),reducestoD2fd(2)2(412+22)3=4g=0: 1:2:4:5 ),onegetstherstordermatchingprior()/(412+22)1=4.Undertheoriginal(1;2)-parameterization,thisisproportionalto(12+22)1=2. 2I0()expfcos(y)g: 22
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Inthesituationwhereinterestliesinseveralparameters,posteriorquantilesarenotwelldenedbutthejointposteriorc.d.f.remainsmeaningfulandprovidesarouteforndingmatchingpriors. 2I11,I=((Ijj0));I1=((Ijj0)),asitsasymptoticvariance,weconsidertherandomvariabley=p 23
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Further,aprior(:)ensuresmatchinginthesamesenseatthesecondorderifandonlyifitsatisesthepartialdierentialequations TheaboveresultsareduetoMukerjeeandGhosh,M.(1997).Thetwoapproachesbasedonthequantiles( 1:2:1:4 )andc.d.f.'s( 1:3:1:2 ),leadtothesamerstordermatchingcondition.However,thecorrespondingsecondordermatchingconditions( 1:2:1:5 )and( 1:3:1:3 )arenotidentical.Thesecondorderconditions( 1:3:1:3 )aremorerestrictivethan( 1:2:1:5 )andoftendonothaveasolution. @1(I11())pXs=2pXv=2@ @sfE@3logf @12@sI11Isv()gpXs=2pXv=2@ @1fE@3logf @1@s@vI11Isv()g=0:(1.3.2.1) and @sfE@3logf @12@sI11Isv()g=0:(1.3.2.2) 24
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Letrg()=(D1g();:::;Dpg())Tbethegradientvectoroftheparametricfunctiong().Denethevector()=(1;:::;p)Tby=[frg()gTI1frg()g]1=2I1rg(): 222),thepopulationmean.HereI11=22;I22=222;I12=0: 1:3:2:3 ),thesolutionobtainedis()=221+1 2221=2: 25
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foralland.WenowgiveacharacterizationforHPDmatchingpriorswheninterestliesintheentireparametervector.TheresultisduetoGhosh,J.K.andMukerjee(1993b)whoreporteditinanotherequivalentform. 26
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1:4:1:1 )becomes()L111I2d df()I1g=constant: 1:2:2:3 ),theaboveisequivalentto TheHPDmatchingcondition( 1:4:2:1 ),arisingforp=1,wasrstreportedinPeers(1968)andisequivalenttothecorrespondingconditiongiveninSeverini(1991). Continuingwithp=1andagainusing( 1:2:2:3 ),aprioroftheform()/Ir,whererisarealnumber,satises( 1:4:2:1 )ifandonlyif Inparticular,takingr=1=2intheabove,( 1:4:2:1 )holdsforJereys'priorifandonlyif Thecondition( 1:4:2:3 )holdsfortheoneparameterlocationandscalemodelsintroducedin( 1:2:2:6 )and( 1:2:2:7 )respectively.Forthesemodels,JereyspriorisHPDmatchingfor.However,evenwithp=1,Jereys'priordoesnotalwaysenjoytheHPDmatchingproperty. 2L111=2(3+2) (12)3: 1:4:2:3 )doesnotholdbut( 1:4:2:2 )issatisedbyr=-1.HenceJereys'priorisnotHPDmatchingforbut()/I1enjoysthisproperty. 27
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Itcanbeveriedthatformodelswhere underorthogonalparameterization,anysecondordermatchingpriorforposteriorquantilesof1isalsoHPDmatchingfor1.HPDmatchingpriorsareinvariantoftheparameterizationaslongastheobjectofinterest,viewedeitherasaparametricfunctionunderanoriginalparameterizationorasacanonicalparameterafterreparameterization,remainsunaltered. 3:1:1 )isnotsatised.Inaddition,norstordermatchingpriorforposteriorquantilesof1isHPDmatchingfor1.SolutionstotheHPDmatchingcondition( 1:4:3:1 )for1
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1:4:3:1 )ifandonlyifoneofthefollowingholds:(a)r1=0;r2=6;r3=3=2;(b)r1=1;r2=13;r3=2;(c)r1=0;r2=1;r3=1;(d)r1=1;r2=2;r3=1=2: 1:4:1:1 )andisHPDmatchingfor=(1;2)T. 29
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1.5.1.1Introduction holds. TheaboveresultisduetoGhosh,J.K.andMukerjee(1991). 1:5:1:1 )becomes()L111I2d df()I1g+2I1d()
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1:2:2:3 ),theaboveisequivalentto Equation( 1:5:1:2 )isinagreementwiththendingsofSeverini(1991)whostudiedthisproblemforscalar. Continuingwithp=1andagainusing( 1:2:2:3 ),aprioroftheform()/Ir,whererisarealnumber,satises( 1:5:1:2 )ifandonlyif Inparticular,takingr=1=2intheaboveandusingtheregularitycondition( 1:2:2:4 ),itfollowsthatJereys'priorsatises( 1:5:1:2 )ifandonlyif Theabove,byTheorem1.2.2,isalsotheconditionunderwhichJereys'priorissecondordermatchingfortheposteriorquantilesof. Thecondition( 1:5:1:4 )holdsfortheoneparameterlocationandscalemodelsintroducedin( 1:2:2:6 )and( 1:2:2:7 )respectively.Forthesemodels,JereyspriorisLRmatchingfor.OntheotherhandforthebivariatenormalmodelconsideredinExamples(1.2.1)and(1.4.1),thecondition( 1:5:1:4 )isnotmetbut( 1:5:1:3 )holdswithr=1.Thusforthismodel()/IisLRmatchingforthoughJerey'spriordoesnotenjoythisproperty. 1:5:1:1 )givenaboveallowstobepossiblymultidimensionalbutpresumesthatnuisanceparametersareabsent.SeveralresultsoncharacterizationsofLRmatchingpriorsinthepresenceofnuisanceparametershavebeenreportedintheliterature.DiCiccioandStern(1994)allowedboththeinterestandthenuisanceparameterstobepossiblymultidimensionalandmadenoassumptionaboutorthogonal 31
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ForageneralcomparisonbetweentheLRandHPDmatchingconditions( 1:5:1:5 )and( 1:4:3:1 )for1,itcanbeshownthatthedierenceinthelefthandsideof( 1:5:1:5 )and( 1:4:3:1 )revealsthatanHPDmatchingpriorsatisfying( 1:4:3:1 )isalsoLRmatchingfor1ifandonlyifitsatises 1.5.2.1Introduction
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and AsnotedinRaoandMukerjee(1995),theclassesofmatchingpriorsbasedonRao'sscoreandWald'sstatisticareidentical.Lee(1989)alsostudiedthematchingproblemassociatedwithWald'sstatistic. Equations( 1:5:2:1 )and( 1:5:2:2 )adduptothematchingcondition( 1:5:1:1 )fortheLRstatistic.Therefore,anymatchingpriorarisingfromRao'sscoreorWald'sstatisticalsoenjoysthesamepropertyfortheLRstatistic.Theconverseishowever,nottrueingeneral. 33
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1:2:2:3 ),thematchingconditions( 1:5:2:1 )and( 1:5:2:2 )reduceto and respectively.By( 1:2:2:3 ),theseconditionsaremetbyJereys'priorifandonlyif 1:5:2:5 )entailsthecorrespondingconditionfortheLRstatistic.Fortheoneparameterlocationandscalemodels,( 1:2:2:3 )againholds.Thusinthesesituations,Jereys'priorenjoysthematchingpropertyforboththescorestatisticandtheWaldstatistic.Ontheotherhand,inExample(1.4.1),concerningabivariatenormalmodelwithunknowncorrelationcooecient,notonly( 1:5:2:5 )failstoholdbutalsonosolutiontothematchingconditions( 1:5:2:3 )and( 1:5:2:4 )isavailable. Table1-1. SimulatedTailProbabilitiesofPosteriorDistributionsinFisher-VonMisesP(;)(0:05;) nJ(2)(;) 50.06050.0518100.05640.0538 Table1-2. SimulatedTailProbabilitiesofPosteriorDistributionsinFisher-VonMisesP(;)(0:95;) nJ(2)(;) 50.94890.9570100.94210.9475 34
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Inthelastchapterwehaveseenthatmatchingisaccomplishedthrougheither(a)posteriorquantiles,(b)distributionfunctions,(c)highestposteriordensity(HPD)regions,or(d)inversionofcertainteststatistics.However,priorsbasedon(a),(b),(c),or(d)neednotalwaysbeidentical.Specically,itmaysohappenthattheredoesnotexistanypriorsatisfyingallthefourcriteria. Inthischapter,weconsiderthebivariatenormaldistributionwheretheparametersofinterestareeitherthe(i)regressioncoecient,(ii)thegeneralizedvariance,i.e.thedeterminantofthevariance-covariancematrix,and(iii)ratiooftheconditionalvarianceofonevariablegiventheotherdividedbythemarginalvarianceoftheothervariable.Wehavebeenabletondapriorwhichmeetsallthefourmatchingcriteriaforeveryoneoftheseparameters. Withthisreparameterization,thebivariatenormaldistributioncanberewrittenas 2X22(X11)2 ItmaybenotedthatistheregressioncoecientofX2onX1,while2=1222(12)isthedeterminantofthevariance-covariancematrix.Also,2=V(X2jX1)=V(X1). 35
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whereA=0B@2 1 Theinverseoftheinformationmatrixisthen whereA1=0B@1 3 32 31CA: 0(1;2;;;)/(;;);(2.1.5) andndsuchthatthematchingcriteriagivenin(a)-(d)areallsatisedfor;andeachindividually.Thiswearegoingtoexploreinthenextfoursections. Beforeendingthissectionwestatealemmawhichisrepeatedlyinthesequel. 2.1.2 ), 36
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andE(X22(X11))2=22+221212=22+2222222=22(12)=: 2.1.6 ).E@logf @3=E(X11)3(X22(X11))3 andE@logf @@2logf @2=E(X11)3(X22(X11)) @2=(X11)2 2.1.7 ).FurtherE@3logf @2@=E(X11)2 2=(2)1;
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@2@=E(X11)2 2=3: 2.1.8 )wenotethatE@3logf @@2=2E(X11)(X22(X11)) @@2=2E(X11)(X22(X11)) 2.1.9 )holds,sometediousalgebraneedstobedone.Tothisendnotethat@logf @3=1 86fX22(X11)g6 24fX22(X11)g2 45fX22(X11)g4
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@3=1 @@2logf @2=1 22fX22(X11)g2 24fX22(X11)g2 25fX22(X11)g2 @@2logf @2=1 4+ 482 2.1.10 )holdsbecauseE@3logf @3=E2 4=4=3;E@3logf @2@=EfX22(X11)g2 32 3=0;
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@@2=EfX22(X11)g2 23=(2)1: 2.1.11 ),wenotethat@logf @3=fX22(X11)g2 @3=15 8315 839 83+9 83=0: @@2logf @2=fX22(X11)g2 @@2logf @2=EfX22(X11)g2 23=(3)1;
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@3=E3fX22(X11)g2 4=33: 1.2.3.4 )itfollowsthattheclassofrstorderprobabilitymatchingpriorsforisgivenby whereg0isanarbitrarysmoothfunctionof(;).Inorderthatsuchapriorsatisesthesecondordermatchingproperty,weneedtondg0bysolving(see( 1.2.3.5 )) @(g0(;)2E@3logf @2@)+@ @(g0(;)2E@3logf @2@)+1 6g0(;)@ @(3E@logf @3)=0:(2.2.2) From( 2.1.6 )and( 2.1.7 )inLemma2.1,( 2.2.2 )reducesto @[g0(;)(=)]+@ @g0(;)=0;(2.2.3) andasolutionto( 2.2.3 )isprovidedbyg0(;)/1.Thustheprior(;;)=11satisesthesecondordermatchingproperty.Nextweproceedtowardsndingasecondordermatchingpriorfor.First,from( 1.2.3.4 ),weobtaintheclassofrstordermatchingpriorsforasgivenby 41
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@g1(;)2E[(X11)(X22(X11))] @(g1(;)2E3(X22(X11))2 6g1(;)@ @3E(@logf @)3=0:(2.2.5) SinceE[(X11)(X22(X11))]=0,E[(X22(X11))2]=andE[(X11)2]==,from( 2.1.9 )ofLemma2.1and( 2.2.5 ),oneneedstosolve2@ @[g1(;)(=)]=0.Anyg1(;)/1g()providesthesolution.Inparticular,takingg=1,(;;)/()1isasecondordermatchingpriorfor. Finally,whenistheparameterofinterest,from( 1.2.3.4 ),onceagain,theclassofrstordermatchingpriorsisgivenby Inordertondasecondordermatchingpriorfor,weneedtosolve @g2(;)122E(@3logf @@2)@2 3@ @g2(;)14E(@3logf @3)=0:(2.2.7) From( 2.1.10 )and( 2.1.11 )ofLemma2.1,( 2.2.7 )reducesto @g2(;)1221 3@ @g2(;)141 Hence,asolutionto( 2.2.8 )isprovidedbyg2(;)/1.Sotheprior(;;)=11satisesthesecondordermatchingpropertyinthiscasetoo.Thusasecondorderquantilematchingpriorwhichworksforevery;andisgivenby(1;2;;;)/()1.Backtotheoriginalparameterization,namely,(1;2;1;2;),thisreducestotheprior 42
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@22E@3logf @2@+2E@3logf @2@(;;)!=0;(2.3.1) and @2E@3logf @@22+E@3logf @@22!=0:(2.3.2) Using( 2.1.7 )and( 2.1.8 )ofLemma2.1,( 2.3.1 )simpliesto@ @[(+)(;;)]=0whichholdstriviallyforanyprior(;;)whichdoesnotdependon,includingtheprior(;;)/()1,theonefoundintheprevioussubsection.Again,withastheparameterofinterest,foranyprior(;;)whichdoesnotdependon,wesolve@ @f(2)122(;;)g+@ @f322(;;)g=@ @(;;)+@ @(;;)=0: 2.3.1 ).Forthisprior( 2.3.2 )reducesto22@ @E@3logf @@2+4@ @E@3logf @@2=0:
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2.1.8 )ofLemma2.1,E@3logf @@2=E@3logf @@2=0.Hencematchingviadistributionsisachievedwithanyprioroftheform(1;2;;;)/h(;),andinparticularh(;)/()1. Nextwhenistheparameterofinterest,forndingamatchingprior,oneneedstosolverst @2@ @!@ @4E@3logf @3!@ @4E@3logf @3!=0;(2.3.3) whichsimpliesto@2 @2@ @!12@ @!=0: @4E@3logf @3=0:(2.3.4) Finallywhenistheparameterofinterest,forndingamatchingprior,oneneedstosolve @2@ @@ @E(@3logf @@2)22@ @E(@3logf @@2)22=0;(2.3.5) and @4E@3logf @3=0:(2.3.6) Again,using( 2.1.7 ),( 2.1.10 )and( 2.1.11 )ofLemma2.1,theprior(;;)=()1satisesthesecondordermatchingproperty. 44
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1.4.3.1 )ofChapter1) @22E@3logf @2@!+@ @4E@3logf @2@!+@ @4E@3logf @3!@2 Using( 2.1.7 )ofLemma2.1,( 2.4.1 )reducesto @()+@ @()@2 Clearlytheprior(;;)/()1satises( 2.4.2 ). Nextconsiderastheparameterofinterest.Nowoneneedstosolve @22E@3logf @2@!+@ @22E@3logf @2@!+@ @4E@3logf @3!@2 Againfrom( 2.1.8 )and( 2.1.10 )ofLemma2.1,( 2.4.3 )simpliesto @()@2 whichissatisedbytheprior(;;)/()1. Finally,whenistheparameterofinterest,weneedtosolve @4E@3logf @2@!+@ @22E@3logf @2@!+@ @4E@3logf @3!@2 From( 2.1.8 ),( 2.1.10 )and( 2.1.11 )ofLemma2.1,( 2.4.5 )reducesto @()+@ @()@2 Again(;;)/()1willdo. 45
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@22(2)1!+@ @43!+@ @2@ @2E@logf @@2logf @22E@3logf @@22E@3logf @@2!=0:(2.5.1) From( 2.1.6 )and( 2.1.8 )ofLemma2.1,( 2.5.1 )reducesto@ @()+@ @()+2@ @(@ @)=0; @()+@ @()+2@2 @2=0: Nextifistheparameterofinterest,LRmatchingpriorforisobtainedbysolvingthedierentialequation @(22:0:)+@ @(22:0:)+@ @2@ @2E@logf @@2logf @22E@3logf @2@+E@3logf @2@!=0:(2.5.2) Againfrom( 2.1.7 ),( 2.1.9 )and( 2.1.10 )inLemma2.1,( 2.5.2 )reducesto@ @[2f@ @+2
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@[2@ @+4]=0 whichholdsfor/()1. Finally,whenistheparameterofinterest,theLRmatchingpriorisobtainedby @(f43:0g)+@ @(221 @2@ @2E@logf @@2logf @22= 20!=0:(2.5.3) Onceagain,using( 2.1.11 )inLemma2.1,( 2.5.3 )reducesto@ @()+@ @2@ @+2 Firstwederivetheposteriorpdfof.Itturnsouttobeapropert-density.Thisimmediatelyimpliestheproprietyofthejointposterioralso,becauseotherwisethemarginalposteriorofcannotbeproper. Tothisend,rstwritingXi=(X1i;X2i)T,i=1;:::;n,thejointposteriorisgivenby 2nXi=1X2i2(X1i1)2 FromtheidentitiesnX1X2i2(X1i1)2=nXi=1f(X2iX2)(X1iX1)2g+n(X22(X11))2
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2.6.1 )aftersimplication 2nXi=1X2iX2X1iX12 Nextintegratingoutin( 2.6.2 )andwritingSjk=Pni=1(XjiXj)(XkiXk);j;k=1;2, From( 2.6.3 ),themarginalposteriorofisgivenby(jX1;X2)/Z1on22+S22+2S112S12 2;(2.6.4) whereS22:1=S22S122=S11.Thisposteriorisat-distributionwithlocationparameterS12=S11,scaleparameterfS22:1=(n2)g1=2anddegreesoffreedomn-2. Nextwendtheposteriorof.Integratingoutin( 2.6.2 ),onegets 21 2exp1 2S22:1 Nowtheposteriorofisgivenby 2Z101 2exp1 2(S22:1 Putting=z1,( 2.6.6 )isrewrittenas 2Z10z3 2expS22:1 48
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2.6.7 )that sothat1hasaGammadistributionwithshapeparametern-2andscaleparameter(S11S22:1)1 2. Finally,integratingin( 2.6.5 ),themarginalposteriorofisgivenby TheconstructionofHPDcredibleintervalsisfairlysimple.Theposteriorofbeingaunivariate-t(thussymmetricandunimodal),from( 2.6.4 ),the100(1)%HPDcredibleintervalforisgivenbyS12=S11fS22:1=(n2)g1=2tn2;=2,wheretn2;=2denotestheupper100 Nextobservingthattheposteriorofislog-concave,the100(1)%regionforisgivenby[1;2],where1and2satisfy and 2d=1:(2.6.11) Ifw=1,thentheposteriorpdfofwisgivenby(wjX1;X2)/wn3exp(wS1=211S1=222:1):
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and 2dw=1:(2.6.13) Clearlythesolution[w1;w2]of( 2.6.12 )and( 2.6.13 )isdierentfromthesolution[12;11]of( 2.6.10 )and( 2.6.11 ). Finallyobservingthattheposteriorofin( 2.6.9 )islog-concave,the100(1)%HPDinterval[1;2]forisobtainedbysolvingn21(21+S22:1 NowweevaluatethefrequentistcoverageprobabilitybyinvestigatingtheHPDcredibleintervalofthemarginalposteriordensitiesof,andunderourprobabilitymatchingpriorforseveralandn.Thatistosay,thefrequentistcoverageofa100(1)%HPDintervalshouldbecloseto1.Thisisdonenumerically.Table2-1givesnumericalvaluesofthefrequentistcoverageprobabilitesof95%HPDintervalsfor,and. Thecomputationofthesenumericalvaluesisbasedonsimulation.Inparticular,forxed(1;2;21;22;)andn,wetake5;000independentrandomsamplesof(X1;X2)fromthebivariatenormalmodel.Inoursimulationstudy,wetake1=2=0withoutlossofgenerality.Undertheprior,thefrequentistcoverageprobabilitycanbeestimatedbytherelativefrequencyofHPDintervalscontainingtrueparametervalue.AninspectionofTable2-1revealsthattheagreementbetweenthefrequentistandposteriorcoverage 50
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Table2-1. FrequentistCoverageProbabilitiesof95%HPDIntervalsfor,andwhen21=1and22=1 n 4 0.9520.9470.949 8 0.9460.9550.950 12 0.9540.9520.948 16 0.9520.9540.950 20 0.9450.9480.950 0.50 4 0.9500.9520.949 8 0.9440.9520.948 12 0.9540.9530.944 16 0.9460.9500.949 20 0.9520.9480.949 0.75 4 0.9550.9520.953 8 0.9530.9480.949 12 0.9500.9460.947 16 0.9480.9460.951 20 0.9560.9460.951 51
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Oneoftheclassicproblemsinstatisticsisinferenceforthecorrelationcoecient,,inabivariatenormaldistribution.BeginningwithFisher'shyperbolictangenttransformation,therehavebeenmanyproposals,bothfrequentistandBayesian,whichaddressthisproblem.AddedtothisistheducialapproachasfoundinFisher(1930,1956)andPratt(1963). BayesianinferenceforbeganintheearlysixtieswiththeworkofBrillinger(1962)andGeiserandCorneld(1963).ThemainobjectiveoftheseauthorswastondwhethertheducialdistributionofcouldbeidentiedasaBayesianposteriorunderpossiblysomedefaultorobjectiveprior,andtheconclusionwasthatthiswasmostlikelynotpossible. Afteralongfallowperiod,interestinthisproblemrevivedwiththerecentinterestingworkofBergerandSun(2006).Theseauthorsconsideredvariousparametricfunctionsarisingfromthebivariatenormaldistribution,andderivedmanyobjectivepriorswhichsatisfythequantilematchingproperty.Intheprocess,theyfoundapriorwhichachievesthisgoal.Inaddition,theyshowedthattheresultingposteriormatchedtheducialdistributionofasproposedbyBrillinger(1962)andGeiserandCorneld(1963). Inthischapter,asbefore,werstconstructanorthogonalparameterizationwithastheparameterofinterestandthenndaprior,ifany,whichmeetsallthematchingcriterion,atleastasymptotically.Inaddition,wehaveconsideredseverallikelihood-basedmethodsaswellforsimilarinferentialpurposesbasedoncertainmodicationsoftheprolelikelihood,namelyconditionalprolelikelihood,adjustedprolelikelihoodandintegratedlikelihood. 52
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thebivariatenormalpdfcanberewrittenas 2(12)1=22(X11)2 Withthisreparameterization,theFisherInformationmatrixreducesto whereA=0B@1 2(12)1=2 2(12)1=21 (12)2): where and 53
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3.1.2 ), @2@1=0;E@3logf @2@2=1 @3=6 @@2logf @2=2 @@21= 21(12)2;E@3logf @@22=0;(3.1.10) 3.1.7 ).E@3logf @2@1=1 221+22 221+22 @2@2=E2(X11)(X22) 2221+22 22(12)3=22 2221+22
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3.1.8 )holdsbecauseE@3logf @3=E2(X11)(X22) 229+63 229+63 (12)33(3+22) (12)3=6 3.1.9 )holds,notethat,usingtheBartlettIdentitywegetE@logf @@2logf @2=d dI+E@3logf @3=d d1 (12)26 3.1.10 )holds,wehaveE@3logf @@21=E(X11)2 21(12)2; @@22=E+2(X11)(X22) 32(12)3=22 55
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1.2.3.4 ),theclassofrstordermatchingpriorsischaracterizedby Asisoftencustomarytoassignauniformpriorto(1;2)onR2,wewillconsideronlythesubclassofpriorswhereg0(1;2;1;2)=g(1;2). Aprioroftheform/(12)1g(1;2)satisesthesecond-orderprobabilitymatchingpropertyifandonlyif(see( 1.2.3.5 ))gsatisestherelation @1g(12)I11E@3logf @2@1+I22E@3logf @2@2+g @(12)3E@3logf @3g@2 Nowby( 3.1.7 )and( 3.1.8 )fromLemma3.1and( 3.1.6 ),( 4.2.2 )simpliesto(12)1@ @2(g2)2g@ @g@2 @2(g2)=0andasolutionisprovidedbyg(1;2)/h(1)12.Thuseveryprior(1;2;1;2;)/h(1)12(12)1isasecondorderprobabilitymatchingpriorforforanyarbitrarysmoothfunctionhof1.Inparticularifweleth(1)=11,thenfromTheorem1ofDattaandGhosh,M.(1995),theone-at-a-timereferenceorreversereferencepriorforisgivenby1112(12)1.ThispriorwasrstproposedinLindley(1965),andwassubsequentlyshowntobeaone-at-a-timereferencepriorbyBayarri(1981).Duetotheinvariancepropertyofsuchaprior,backtotheoriginalparameterization,asecondordermatchingpriorforis(1;2;1;2;)/1112(12)1. Therstorderquantilematchingprior/(12)1g(1;2)isalsorstordermatchingviadistributionfunctions.Itfollowsfrom( 1.3.2.1 )and( 1.3.2.2 )ofChapter1thatinorderthatthispriorisalsoasecondorderdistributionfunctionmatchingprior,it 56
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@fI@ @g@ @2fE@3logf @2@2II22g@ @fE@3logf @@21II11g=0(3.2.3) and @fE@3logf @3(I)2g=0:(3.2.4) Itiseasilyveriedthatfortheprior/(12)1g(1;2),thelefthandsideof( 3.2.4 )reducesto6g.Theaboveprioralsofailstosatisfy( 3.2.3 )foranyg.Hencewedonothaveapriorthatsatisesthesecondorderdistributionfunctionmatchingcriteria. 1.4.3.1 )ofChapter1)weneedapriorwhichsatisesthedierentialequation @1(12)2I11E@3logf @2@1+@ @2(12)2I22E@3logf @2@2+@ @(12)4E@3logf @3@2 Using( 3.1.7 )and( 3.1.8 )fromLemma3.1and( 3.1.6 ),( 3.3.1 )reducesto @226@ @(12)@2 Considertheclassofpriors(1;2;)/h(1)a2(12)b.Withthisprior,( 3.3.2 )canbewrittenash(1)a2[(a+1)(12)b+2(b1)f(12)b+12(b+1)2(12)bg]=0: 57
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Hencethetwopossiblesolutionsarea=1;b=1anda=4;b=3=2.Thisresultsin/h(1)12(12)and/h(1)42(12)3=2whicharebothHPDmatchingfor.Inparticularforh(1)=11,backtotheoriginalparameterization,weobtain/1112(12)and/4142(12)asHPDmatchingfor.Ingeneral,HPDmatchingpriorssuerfromlackofinvariance.However,ifthesameobjectofinterestisconsideredoverthetwoparameterizationsthentheyareinvariantoftheparameterizationadopted.ThishasbeendiscussedindetailinDattaandMukerjee(2004,p74). 1.5.1.5 ),alikelihoodratiomatchingpriorisobtainedbysolving @u()I111IsuE(@3logf @21@s)+@ @1I111@ @1)(I111E((@logf @1)(@2logf @21))pXs=2pXu=2IsuE(@3logf @1@u@s)=0:(3.4.1) Undertheorthogonalparameterizationobtainedin( 3.1.1 ),( 3.4.1 )canberewrittenas @1(12)2I11E@3logf @2@1+@ @2(12)2I22E@3logf @2@2+@ @(12)2@ @(12)2E(@logf @)(@2logf @2)I11E@3logf @@21I22E@3logf @@22!=0:(3.4.2) 58
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3.1.7 ),( 3.1.9 )and( 3.1.10 )fromLemma3.1and( 3.1.6 ),( 3.4.2 )reducesto @22+@ @(12)2@ @3 Consideragaintheclassofpriors=h(1)a2(12)b.Then( 3.4.3 )furtherreducesto Inorderthat( 3.4.4 )holdsforall(1;1)auniquesolutionisobtainedfora=1andb=3=2.Hencetheuniquepriorwithintheconsideredclassofpriorsthatsatisesthelikelihoodratiomatchingpropertyisgivenby/h(1)12(12)3=2.Onceagain,ifweleth(1)=11,thenbacktotheoriginalparameterization,/1112(12)3=2satisesthelikelihoodratiomatchingpropertyfor. 2(12)1=22nXi=1(X1i1)2 Nextconsiderthetransformation=;21=12
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2(12)nXi=1(X1i1)2 Now,integratingout1and2,weobtain 2(12)n1 2+aexp1 2(12)S11 Consideranothertransformationz1=21(12);z2=22(12)andz3=: 2+a(z1z2)n+1 2exp1 2S11 2+a(z1z2)n+1 2exp1 2S11 2+a1Xr=0Z10Z10(z1z2)n+r+1 2Sr12 2fS11 2+a1Xr=0Sr12zr3 21exp1 2fw1S11+w2S22gdw1dw2=(1z23)n1 2+a1Xr=0zr3Rr 2):(3.5.4) 60
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2+adz3=0.Soinordertoshowthepropriety,weneedtoonlyshowthatI=Z11(1z23)n1 2+a1Xr=0z2r3R2r 2)dz3<1: 2+adz3=2Z10z2r3(1z23)n1 2+adz3=Z10ur1=2(1u)n1 2+adu=Beta(r+1 2;n+1 2+a)=(r+1 2)(n+1 2+a) (n+2r+2 2+a)fora>n2 2: 2)2(n+2r1 2) (n+2r+2 2+a):(3.5.5) BytheLegendreduplicationformula,(2r)!=(2r+1)=(r+1 2)(r+1)22r=1=2.Hencewritingk(>0)asagenericconstantwhichdoesnotdependonr,I=k1Xr=0R2r 2) (r+n+2+2a 2)2 61
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Webeginwiththeprolelikelihoodfor(1;2;)givenby 2(12)1=22S21 whereS21=Pni=1(X1iX1)2;S22=Pni=1(X2iX2)2.LetlplogLp. Toobtainthemaximizeroftheprolelikelihoodrstweobtain@lp 2(12)1=22S21 ^1()=s
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2(12)1=22S21 ^2()=S1S2(1r) Thustheprolelikelihoodforisgivenby (1r)n(12)n=2/(12)n=2 Next,from( 3.1.3 )thedeterminantofthematricesAandBare1 2 Nextwederivetheadjustedprolelikelihoodfor.LetdenotethevectorofnuisanceparametersandU()thescorefunction,thatisU()=dlogLp() dm()o=var;^(U()).Alsolet~U()=fU()m()gw().Theadjustedproleloglikelihoodforisobtainedaslap()=R~U(t)dt.From( 3.6.3 ),thescorefunctionisthengivenby (1r)(12):(3.6.5) FromKendallandStuart(Vol1,1968,pg390)and( 3.6.3 ) and 63
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12g2=n2 12+32(r)2 Hence Further,d2lp() dfr d1 (1r)1 (12)g=nd d1 (12)2=n2r2(12r+2r2) 12g2+132
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Thisleadsto var[U()]=n Thus ~U()=[U()m()]w()=[U() Inotherwordsdlap() 4log(12): 4(1r)n:(3.6.13) Finallywewishtondtheintegratedlikelihood.Thisrequiresspecicationofapriordistributionforthenuisanceparametersconditionalontheparameterofinterest.Inparticular,theconditionalreferencepriorofBerger,LiseoandWolpert(1999)isgivenby(1;2;1;2;)/1122(12)1=2.Thencalculationssimilartothosedonefor 65
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21Xa=0ara OnecommonfeatureofallthemodiedlikelihoodsLP();LCP();Lap()andLI()isthattheyarealldependentonthedataonlythroughthesamplecorrelationcoecientr. Prior1/(12)1112Prior2/(12)3=21112Prior3/(12)11112. Sincethefullconditionaldistributionoftheparametersunderanyofthethreepriorsdonotfollowastandarddistributionalform,weusedGibbssamplingwithcomponentwiseMetropolis-Hastingsupdatesateachiterationtogeneraterandomnumbersfromtheconditionalposteriordistributionsofeachparameter(RobertandCasella,2001).Werantwochainswithdierentinitialvaluesandallowedaburn-inof4000each.Arandom-walkjumpingdensitywithnormalnoiseaddedtotheexistingvalueinthechainforthemeansandlogstandarddeviationswereused.Thecorrelationsalsohadarandomwalkpriorbyaddingasmallnormalnoisetotheoldvalues.Eachchainwasrun10,000timesandconvergencewasjudgedbyaGelman-Rubin(GelmanandRubin,1992)diagnostic.Thetraceplotpresentingthetimehistoryofall10000iterationsforallveparametersispresentedforasamplesimulateddatasetwith=0:3,underPrior3andsamplesize10,inFigure3-1.Figure3-2presentstheplotofGelman-Rubindiagnosticforthechainunderthesamesettingwithdiagnosticvaluescloseto1suggestingconvergence.Figures3-3,3-4and3-5areposteriordistributionsforunderthreedierentpriorsforfourdierentsamplesizesn=10;20;30;40.Onecanimmediatelymakethefollowing 66
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WerepeatedourGibbssamplingestimationtechniquefor500datasetsundereachcongurationofandn.Eachtime,wecomputedtheposteriormean,the95%quantileinterval(asgivenbythe2.5thand97.5thsamplepercentileoftherandomlygeneratedparametervaluesaftertheburn-inperiod)andthe95%HPDinterval.Table3-1presentstheaverageofposteriormeans,themeansquarederror,thefrequentistcoverageoftheBayesiancredibleintervals(asestimatedbytheproportionoftimesthetrueparametervaluefallsinthecorrespondingcredibleintervals)acrossthe500datasetsandunderthreedierentpriors.Someinterestingdierencescanbenotedinthebehaviorforsmallersamplesizes.Prior1appearstobeperformingworsethanPriors2and3whereaspointestimationofisconcerned,withhigherbias,thoughtheMSEisnotnecessarilylargerforallvaluesof.Ontheotherhand,Prior1hasappreciablybettercoveragepropertyfortheHPDintervalsforsmallersizes,thanPriors2and3,andisinfactthetheoreticallyestablishedHPDmatchingprior.Priors1and3areverycomparableintermsofcoverageofquantileintervals,withPrior3havingaslightedgeoverPrior1asitattainsnominalcoverageforasmallerninmanycases.Prior2,elicitedfromainversionoflikelihood-ratiostatisticpointofviewappearstobetheleastattractivefromfrequentistcoverageperspective.Basedonoursimulationresults,ifoneisconcernedaboutboth 67
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Figure3-1. PlotofGelman-RubinDiagnosticStatisticforUnderPriorIIIforn=10UndertheSimulationSettingofSection3.7. 68
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SimulationResulttoComparetheThreeDierentPriorsSuggestedforBivariateNormalCorrelationParameter.TheTrueParameterSettingsare1=2=0,1=2=1andVaryingValuesofasListed.Prior1:/(12)1112,Prior2:/(12)3=21112,Prior3:/(12)11112.ResultsareBasedon500SimulatedDatasets.:Averagevalueforposteriormeanof,averagedacross500simulateddatasets. 10-0.650.050.870.91-0.800.020.860.82-0.770.020.880.86-0.820-0.710.020.870.90-0.780.010.900.89-0.770.010.900.9030-0.740.010.900.92-0.790.010.920.91-0.780.010.930.9240-0.760.010.940.95-0.790.000.930.91-0.790.000.940.92 10-0.350.070.920.92-0.470.080.900.87-0.440.070.920.89-0.520-0.410.040.910.91-0.480.040.910.90-0.460.030.910.9130-0.440.020.940.93-0.490.020.940.93-0.480.020.950.9340-0.440.020.950.96-0.480.010.960.94-0.470.010.970.95 10-0.140.060.940.92-0.180.100.90.85-0.170.090.910.87-0.220-0.160.030.940.96-0.200.050.940.90-0.190.040.950.9230-0.170.030.940.93-0.190.030.920.91-0.190.030.930.9240-0.170.020.950.94-0.190.030.940.92-0.180.020.940.94 100.010.060.960.920.010.110.890.860.010.090.910.880200.020.040.960.930.020.050.930.900.020.050.930.9030-0.010.020.930.92-0.010.030.920.90-0.010.030.910.904000.020.960.950.000.030.950.920.000.030.940.94 100.150.060.940.930.210.100.900.850.190.090.930.880.2200.150.040.950.930.180.050.930.890.170.050.930.90300.170.030.940.930.200.030.930.920.190.030.940.92400.170.020.950.940.190.020.930.920.180.020.930.93 100.350.070.930.910.480.080.880.840.450.070.920.870.5200.410.030.930.920.480.030.930.900.470.030.930.92300.440.020.930.930.490.020.920.920.480.020.920.92400.450.020.950.950.490.020.960.940.480.020.950.95 100.630.060.840.890.780.030.860.820.740.030.890.860.8200.710.020.890.920.790.010.930.910.770.010.930.93300.750.010.900.930.800.010.940.920.790.010.950.93400.760.010.920.940.790.000.940.930.790.000.940.94
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SampleTracePlotforAlltheParametersunderPriorIIIforn=10UndertheSimulationSettingofSection3.7 70
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PosteriorDistributionforunderPriorIforDierentSampleSizes,UndertheSimulationSettingofSection3.7 71
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PosteriorDistributionforunderPriorIIforDierentSampleSizes,UndertheSimulationSettingofSection3.7 72
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SamplePosteriorDistributionforunderPriorIIIforDierentSampleSizes,UndertheSimulationSettingofSection3.7 73
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Therearemanyexperimentalsituationsinwhichaninvestigatorwantstoestimatetheratioofvariancesoftwoindependentnormalpopulations.Studyoftheratioofvariancesdatesbackto1920whenFisherdevelopedtheF-statisticfortestingthevarianceratio.Themostwell-usedexampleinvolvestestingofthehypothesisthatthestandarddeviationsoftwonormallydistributedpopulationsareequal.AlthoughratioofvarianceshavebeenvigorouslystudiedinthecaseoftwoindependentnormalsamplesbothinthefrequentistandintheBayesianliterature,littlestudyhasbeendoneforapossiblycorrelatedbivariatenormalpopulation.Fortestingtheequalityofvariancesinabivariatenormalpopulation,Pitman(1939)andMorgan(1939)introducedavariabletransformationwhichreducestheproblemtotestingabivariatenormalcorrelationcoecientequaltozero.Thissameideacanbeeasilyextendedtotestthenullhypothesiswhetheravarianceratioequalsaparticularvalue.Invertingthisteststatistic,RoyandPottho(1958)obtainedcondenceboundsontheratioofvariancesinthecorrelatedbivariatenormaldistribution.SincetheteststatistichasaStudents'st-distributionunderthenullhypothesis,theresultingcondenceboundsinvolvespercentilesofaStudentt-distribution. TheobjectiveofthisChapteristondpriorsaccordingtothedierentmatchingcriteriawhentheratioofvariancesinthebivariatenormaldistributionistheparameterofinterestandcomparetheperformanceformoderatesamplesizes.Itturnsoutthatthereisageneralclassofpriorswhichsatisesallthematchingcriteria. 74
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andobtainthebivariatenormalpdfasin( 3.1.2 ). Withthisreparameterization,theFisherInformationmatrixreducesto whereA=0B@1 2(12)1=2 2(12)1=21 (12)2): Theinverseoftheinformationmatrixissimplythen where and Forsubsequentsections,weneedalsoafewotherresultswhicharecollectedinthefollowinglemma. 75
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3.1.2 ), @21@= 21(12)2;E@3logf @21@2=1 @31=3 @1@2logf @21=1 @1@2=0;E@3logf @1@22=0;(4.1.9) 4.1.6 ).E@3logf @21@=E(X11)2 21(12)2 @21@2=E(X11)2 4.1.7 ),weseethatE@3logf @31=E3(X11)2 4.1.8 )holdsbecausefromtheBartlettIdentityE@logf @1@2logf @21=E@3logf @31@ @11
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4.1.9 )holdsbecauseE@3logf @2@1=1 221+22 221+22 whileE@3logf @1@22=1 (12)1=232(X11)2 (12)1=23212 1.2.3.4 ),theclassofrstordermatchingpriorsischaracterizedby Asisoftencustomarytoassignauniformpriorto(1;2)onR2,wewillconsideronlythesubclassofpriorswhereg0(1;2;;2)=g(;2). Aprioroftheform/11(12)1=2g(;2)satisesthesecond-orderquantilematchingpropertyifandonlyif(see( 1.2.3.5 )ofChapter1)gsatisestherelation @211(12)1=2g2122E@3logf @21@2+@ @11(12)1=2g21(12)2E@3logf @21@+1 6(12)1=2g@ @131(12)3=2E@logf @13=0(4.2.2) From( 4.1.5 )-( 4.1.7 ),( 4.2.2 )simpliesto @2g211@ @g(12)1=2=0:(4.2.3) 77
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2.Withthischoiceofgthelefthandsideoftheaboveequationreducesto 2@ @2a+1211a2@ @jja+1(12)a+3 2+1sgn()=(a+1)11a2jja(12)a+3 211a2jja+1a+1 2(12)a+3 2(2)sgn()+(12)a+3 2+1(a+1)jjasgn2=(a+1)11a2jja(12)a+3 212+(12)=0:(4.2.4) Thuseveryprior(1;2;1;2;)/11a2jja(12)a+3 2isasecondorderprobabilitymatchingpriorfor1.Duetotheinvariancepropertyofsuchaprior,backtotheoriginalparameterization,asecondordermatchingpriorfor1 1.3.2.1 )and( 1.3.2.2 )ofChapter1thatinorderthatthisclassofpriorsalsosatisesthesecondorderdistributionfunctionmatchingcriterion,itneedstosatisfythetwodierentialequations @1(I11@ @1)pXs=2pXv=2@ @sE@3logf @12@sI11Isv()pXs=2pXv=2@ @1E@3logf @1@s@vI11Isv()=0:(4.3.1) and @sE@3logf @12@sI11Isv()=0:(4.3.2) Inourcontext,when1=1 78
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4.3.1 ),( 4.3.2 )and( 4.1.5 ) @121(12)1=2@ @111@ @211g21(12)1=222E@3logf @21@2@ @11g21(12)1=2(12)2E@3logf @21@g@ @1E@3logf @1@2221(12)1=22211g@ @1E@3logf @1@212(12)1=2(12)211=0(4.3.3) and @1E@3logf @311141(12)2=0:(4.3.4) From( 4.1.6 )and( 4.1.9 )ofLemma4.1,( 4.3.3 )reducesto @2g211@ @g(12)1=2=0(4.3.5) whilethelefthandsideof( 4.3.4 )reducesto3g(12)1=2@ @1(12)whichisclearly0foranyg.So,weneedtondgsuchthat( 4.3.5 )issatised.Inparticular,( 4.3.5 )issatisedifweletgonceagaintobetheclassoffunctionsg(2;)=a2jja(12)a+2 2.Inotherwords,thesameclassofpriorsenjoysecondordermatchingforbothquantilesaswellasdistributionfunctions. 3.1.1 )ofChapter1anysecondordermatchingpriorforposteriorquantilesof1isalsoHPDmatchingfor1inthespecialcaseofmodelssatisfying @1I11E@3logf @31=0:(4.5.1) 79
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4.1.5 )and( 4.1.7 ),( 4.5.1 )holdsandhencethesecondorderquantilematchingprior/11a2jja(12)a+3 2isalsoHPDmatching. 1.5.1.5 ),alikelihoodratiomatchingpriorisobtainedbysolving @221(12)22E@3logf @21@2+@ @21(12)(12)2E@3logf @21@+@ @121(12)@ @121(12)E@logf @1@2logf @2122E@3logf @1@22(12)2E@3logf @1@2=0(4.6.1) Thenfrom( 4.1.6 ),( 4.1.8 )and( 4.1.9 )ofLemma4.1,( 4.6.1 )reducesto @22+@ @(12)+@ @121(12)@ @1+11=0(4.6.2) Consideronceagain(1;2;1;2;)/11a2jja(12)a+3 2.Then@ @1+11=0;andthelefthandsideof( 4.6.2 )simpliesto11jja(12)a+3 2@ @2a+1211a2@ @jja+1(12)a+3 2+1 4.2.4 ),andleadstothesameclassofmatchingpriorsasbefore.Withthisweconcludethatwehavebeenabletondaclassofpriors(1;2;1;2;)/a1a2jja(12)1whichsatisesallthedierentmatchingcriteria. 2satisesthevariousmatchingpropertiesdiscussedabove.Also,thejointposterior 80
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2(12)1=22nXi=1(X1i1)2 2:(4.7.1) Nextconsiderthetransformation=;1=(12)1=2 2(12)nXi=1(X1i1)2 Now,integratingout1and2,weobtain 2exp1 2(12)S11 Consideranothertransformationz1=21(12);z2=22(12)andz3=: 2+najz3ja(z1z2)n+a 2S11 2a(z1z2)n+a 2(S11
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2a1Xr=0zr3(2R)r 2)fora0)asagenericconstantwhichdoesnotdependonr,I=k1Xr=0R2r 2)(r+a+1 2) (r+1 2): 2 2(r+na2 2)!R2<1asr!1: 82
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2).Thispriorsatisesquantilematching,matchingviadistributionfunctions,HPDmatchingaswellaslikelihoodratiomatchingproperty. Therearethreepriorsthatwewishtocompare.Therstoneis/11.ThiswasrecommendedbyStaicu(2007)inherPhDdissertationshowingthatthispriorachievesmatchinguptoO(n3=2).Thesecondprioris/11(12)3=2.ThiswassuggestedbyMukerjeeandReid(2001).Thisisaspecialcase(a=0)oftheclassofpriorsthatweobtainedsatisfyingallthematchingcriteria.Finally,theprior/1112(12)1wasrecommendedbyBergerandSun(2007).Thisisalsoone-at-a-timereferencepriorforeachoneoftheparameters1;2andsatisfyingtherstordermatchingproperty. Inordertoevaluatethethreedierentpriors,weundertookasimulationstudywheredatawasgeneratedfromabivariatenormaldistributionwith(1;2;2;)=(0;0;1;0:5)andvaryingvaluesof1andvaryingsamplesizesn.Thevaluesof1variedfrom0.5to2.0. Sincethefullconditionaldistributionoftheparametersunderanyofthethreepriorsdonotfollowastandarddistributionalform,weusedGibbssamplingwithcomponentwiseMetropolis-Hastingsupdatesateachiterationtogeneraterandomnumbersfromtheconditionalposteriordistributionsofeachparameter(RobertandCasella,2001).Werantwochainswithdierentinitialvaluesandallowedaburn-inof10000each.Arandom-walkjumpingdensitywithnormalnoiseaddedtotheexistingvalueinthechainforthemeansandlogstandarddeviationswereused.Thecorrelationsalsohadarandomwalkpriorbyaddingasmallnormalnoisetotheoldvalues.Eachchainwasrun40,000timesandconvergencewasjudgedbyaGelman-Rubin(GelmanandRubin,1992)diagnostic.Thetraceplotpresentingthetimehistoryofthelast8000iterationsforallveparametersispresentedforasamplesimulateddatasetwith1=0:7,underPrior3 83
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WerepeatedourGibbssamplingestimationtechniquefor500datasetsundereachcongurationof1andn.Eachtime,wecomputedtheposteriormean,the95%quantileinterval(asgivenbythe2.5thand97.5thsamplepercentileoftherandomlygeneratedparametervaluesaftertheburn-inperiod)andthe95%HPDinterval.Table4-1presentstheaverageofposteriormeans,themeansquarederror,thefrequentistcoverageoftheBayesiancredibleintervals(asestimatedbytheproportionoftimesthetrueparametervaluefallsinthecorrespondingcredibleintervals)acrossthe500datasetsandunderthreedierentpriors.Someinterestingdierencescanbenotedinthebehaviorforsmallersamplesizes.Prior2appearstobeperformingbestintermsofbothcoverageofquantileandHPDintervalsandalsohasexcellentpointestimationpropertiesintermsofaverageposteriormeanandMSEforsmallersamplesizes.Forlargersamplesizesallthreepriorsbecomealmostindistinguishableintermsoftheirperformances. 84
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SimulationResulttoComparetheThreeDierentPriorsSuggestedforBivariateNormalRatioofStandardDeviationParameter1.TheTrueParameterSettingsare1=2=0,2=1andVaryingValuesof1=1asListed.Prior1:/11,Prior2:/(12)3=211,Prior3:/(12)11112.Resultsarebasedon500simulateddatasets.:Averagevalueforposteriormeanof1,averagedacross500simulateddatasets. 100.520.020.930.940.510.020.950.950.520.020.930.930.5200.510.010.940.940.500.010.950.940.510.010.940.94300.500.010.940.940.500.010.950.960.50.010.940.94400.510.010.960.950.510.010.960.950.510.010.950.95 101.060.140.930.941.060.120.950.951.070.140.930.931.0201.010.040.960.941.010.030.950.951.010.040.950.94301.010.030.960.951.010.020.960.951.010.030.950.95401.010.020.950.941.010.020.950.951.010.020.950.94 101.610.250.920.941.610.230.950.951.610.260.930.941.5201.540.120.940.921.540.110.930.931.540.120.930.93301.510.060.940.941.510.050.930.921.510.060.930.93401.520.050.940.941.520.050.950.951.520.050.950.95 102.110.440.940.942.110.420.950.942.110.420.960.942.0202.040.180.950.952.040.180.950.952.040.180.950.95302.040.110.950.962.040.100.950.952.040.110.940.95402.030.080.950.952.030.070.950.952.030.080.950.94
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SampletraceplotforalltheparametersunderPrior3forn=20underthesimulationsettingofSection4.8,Truevalueof1=0:7. 86
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PlotofGelman-RubinDiagnosticStatisticfor1underPrior3forn=20underthesimulationsettingofSection4.8,Truevalueof1=0:7. 87
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PosteriorDistributionfor1underPrior1forDierentSampleSizes,undertheSimulationSettingofSection4.8.Truevalueof1=0:7. 88
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PosteriorDistributionfor1underPrior2forDierentSampleSizes,undertheSimulationSettingofSection4.8.Truevalueof1=0:7. 89
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SamplePosteriorDistributionfor1underPrior3forDierentSampleSizes,undertheSimulationSettingofSection4.8.Truevalueof1=0:7. 90
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Studyofprobabilitymatchingpriors,thatensureapproximatefrequentistvalidityofposteriorcrediblesets,hasreceivedmuchattentioninrecentyears.Inthisdissertation,wedevelopsomesuchpriorsforparametersandsomefunctionsoftheparametersofabivariatenormaldistribution.ThecriterionusedistheasymptoticmatchingofcoverageprobabilitiesofBayesiancredibleintervalswiththecorrespondingfrequentistcoverageprobabilities.Thepaperusesvariousmatchingcriteria,namely,quantilematching,matchingofdistributionfunctions,highestposteriordensitymatching,andmatchingviainversionofteststatistics.Orthogonalparameterizationswereobtainedwhichsimpliedthedierentialequationsthatneededtobesolvedforobtainingthesematchingpriors. First,weconsideredthe(i)regressioncoecient,(ii)thegeneralizedvariance,i.e.thedeterminantofthevariance-covariancematrix,and(iii)ratiooftheconditionalvarianceofonevariablegiventheotherdividedbythemarginalvarianceoftheothervariableastheparametersofinterest.Herewehavebeenabletondasinglepriorwhichmeetsallthefourmatchingcriteriaforeveryoneoftheseparameters.TheagreementbetweenthefrequentistandposteriorcoverageprobabilitiesofHPDintervalsisquitegoodfortheprobabilitymatchingpriorsevenforsmallsamplesizes. Nextweconsiderthebivariatenormalcorrelationcoecientastheparameterofinterest.Hereweobtaindierentpriorssatisfyingthedierentmatchingcriteriaandcomparetheirperformanceformoderatesamplesizes.Therehowever,doesnotexistapriorthatsatisesthematchingviadistributionfunctionscriterion.Inaddition,wedevelopinferencebasedoncertainmodicationsoftheprolelikelihood,namelyconditionalprolelikelihood,adjustedprolelikelihoodandintegratedlikelihood.Onecommonfeatureofallthemodiedlikelihoodsisthattheyarealldependentonthedataonlythroughthesamplecorrelationcoecientr. 91
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Recently,SunandBerger(2006)haveillustratedobjectiveBayesianinferenceforthemultivariatenormaldistributionusingdierenttypesofformalobjectivepriors,dierentmodesofinferenceanddierentcriteriainvolvedinselectingoptimalobjectivepriors.They,inparticular,focusonreferencepriors,andshowthattheright-Haarpriorisaone-at-timereferencepriorformanyparametersandfunctionsofparameters.Ourfutureresearchwillconcentrateonndingprobabilitymatchingpriorsforthemultivariateanalogsofthebivariatenormalparameters.Hereinterestliesinseveralparametersorparametricfunctions.Forinstance,wemaybeinterestedinthegeneralizedvariance,theregressionmatrixorthecorrelationmatrix.Thenposteriorquantilesarenotwell-dened.HPDregionsandcredibleregionviatheLRstatisticremainmeaningfulandofmuchinterest.Theycanbeusedtondmatchingpriors.Alsothejointposteriorc.d.fremainsmeaningfulandprovidesaviablerouteforndingmatchingpriors.Orthogonalparameterizationsarenotguaranteed.Howeveriffound,theywillsimplifythecomputations. 92
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Bayarri,M.J.(1981),\Inferenciabayesianasobreelcoecientedecorrelacindeunapoblacinnormalbivariante,"inTrabajosdeEstadisticaeInvestigacionOperativa,32,18-31. Bartlett,M.S.(1937),\PropertiesofSuciencyandStatisticalTests",Pro.Roy.Soc.LondonA,160,268-282. Berger,J.,andBernardo,J.M.(1992a),\OntheDevelopmentofReferencePriors"(withdiscussion),inBayesianStatistics4,eds.J.M.Bernardo,J.O.Bereger,A.P.Dawid,andA.F.M.Smith,Oxford,U.K.:OxfordUniversityPress,pp.35-60. Berger,J.,Liseo,B.,andWolpert,R.L.(1999),\IntegratedLikelihoodMethodsforEliminatingNuisanceParameters,"StatisticalScience,14,1-22. Berger,J.,andSun,D.(2007),\ObjectivepriorsfortheBivariateNormalModel,"toappearintheAnnalsofStatistics. Berger,J.,andSun,D.(2006),\ObjectivepriorsforaBivariateNormalModelwithMultivariateGeneralizations,"inISDSTechnicalReport,DukeUniversity. Bernardo,J.M.(1979),\ReferencePosteriorDistributionsforBayesianInference"(withdiscussion),JournaloftheRoyalStatisticalSociety,Ser.B,41,113-147. Cox,D.R.,andReid,N.(1987),\OrthogonalParametersandApproximateConditionalInference"(withdiscussion),JournaloftheRoyalStatisticalSociety,Ser.B,49,1-39. Datta,G.S.,andGhosh,J.K.(1995a),\NoninformativePriorsforMaximalInvariantParameterinGroupModels,"Test,4,95-114. Datta,G.S.,andGhosh,J.K.(1995b),\OnPriorsProvidingFrequentistValidityforBayesianInference,"Biometrika,82,37-45. Datta,G.S.,andGhosh,M.(1995a),\SomeRemarksonNoninformativePriors,"JournaloftheAmericanStatisticalAssociation,90,1357-1363. Datta,G.S.,andGhosh,M.(1996),\OntheInvarianceofNoninformativePriors,"AnnalsofStatistics,24,141-159. Datta,G.S.,Ghosh,M.,andMukerjee,R.(2000),\SomeNewResultsonProbabilityMatchingPriors,"CalcuttaStatisticsAssociationBulletin,50,179-192. Datta,G.S.,andMukerjee,R.(2004),ProbabilityMatchingPriors:HigherOrderAsymptotics,LecturenotesinStatistics,Springer,NewYork. Datta,G.S.,andSweeting,T.J.(2005),\ProbabilityMatchingPriors,"HandbookofStatistics,Vol25:BayesianThinking:ModelingandComputation,eds.D.Dey,andC.R.Rao,Elsevier,pp.91-114. 93
PAGE 94
DiCicio,T.J.,andStern,S.E.(1994),\FrequentistandBaysianBartlettCorrectionofTestStatisticsbasedonAdjustedProleLikelihoods,"JournaloftheRoyalStatisticalSociety,Ser.B,56,397-408. Fisher,R.A.(1956),StatisticalMethodsandScienticInference,OliverandBoyd,Edinburgh. Garvan,C.W.,andGhosh,M.(1997),\NoninformativePriorsforDispersionModels,"Biometrika,84,976-982. Garvan,C.W.,andGhosh,M.(1999),\OnthePropertyofPosteriorsforDispersionModels,"JournalofStatisticalPlanningandInference,78,229-241. Gelman,A.,andRubin,D.B.(1992),\InferencefromIterativeSimulationUsingMultipleSequences,"StatisticalScience,7,457-472. Ghosh,J.K.(1994),HigherOrderAsymptotics,InstituteofMathematicalStatisticsandAmericanStatisticalAssociation,Hayward,California. Ghosh,J.K.,andMukerjee,R.(1991),\CharacterizationofPriorsunderwhichBayesianandFrequentistBartlettCorrectionsareEquivalentintheMultiparameterCase,"JournalofMultivariateAnalysis,38,385-393. Ghosh,J.K.,andMukerjee,R.(1992b),\BayesianandFrequentistBartlettCorrectionsforLikelihoodRatioandConditionalLikelihoodRatioTests,"JournaloftheRoyalStatisticalSociety,Ser.B,54,867-875. Ghosh,J.K.,andMukerjee,R.(1993a),\OnPriorsthatmatchPosteriorandFrequentistDistributionFunctions,"CanadianJournalofStatistics,21,89-96. Ghosh,J.K.,andMukerjee,R.(1993b),\FrequentistValidityofHigherPosteriorDensityRegionsinMultiparameterCase,"AnnalsoftheInstituteofStatisticalMathematics,45,293-302. Ghosh,J.K.,andMukerjee,R.(1994b),\AdjustedversusConditionalLikelihood:PowerPropertiesandBartlett-typeAdjustment,"JournaloftheRoyalStatisticalSociety,Ser.B,56,185-188. Ghosh,J.K.,andMukerjee,R.(1995),\FrequentistValidityofHigherPosteriorDensityRegionsinthepresenceofNuisanceParameters,"StatisticalDecisions,13,131-139. Ghosh,M.,Carlin,B.P.,andSrivastava,M.S.(1995),\ProbabilityMatchingPriorsforLinearCalibration,"Test,4,333-357. 94
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Ghosh,M.,andMukerjee,R.(1998),\RecentDevelopmentsonProbabilityMatchingPriors,"in:S.E.Ahmed,M.AhsanullahandB.K.Sinha,eds.,AppliedStatisticalScience,III,,NovaSciencePublishers,NewYork,pp.227-252. Ghosh,M.,andYang,M.C.(1996),\NoninformativeproblemfortheTwo-SampleNormalproblem,"Test,5,145-157. GhoshM.(2001),\IntervalEstimationforaBinomialProportion:Comment,"StatisticalScience,16,124-125. Godambe,V.P.(1960),\AnOptimumPropertyofRegularMaximumLikelihoodEstimation,"AnnalsofMathematicalStatistics,31,1208-1211. Huzurbazar,V.S.(1950),\ProbabilityDistributionsandOrthogonalParameters,"inProceedingsCambridgePhil.Society,46,281-284. Jereys,H.(1961),TheoryofProbability,Oxford,U.K.:OxfordUniversityPress. Kalbeisch,J.D.,andSprott,D.A.(1970),\ApplicationofLikelihoodMethodstoModelsInvolvingLargenumberofParameters"(withdiscussion),JournaloftheRoyalStatisticalSociety,Ser.B,32,175-208. Kass,R.E.,andWasserman,L.(1996),\TheSelectionofPriorDistributionsbyFormalRules,"JournaloftheAmericanStatisticalAssociation,91,1343-1370. Kendall,M.G.,andStuart,A.(1969),TheAdvancedTheoryofStatistics,Vol1:390,HafnerPublishingCompany,NewYork. Lee,C.B.(1989),ComparisonofFrequentistCoverageProbabilityandBayesianPosteriorCoverageProbability,andApplications.UnpublishedPh.D.dissertation,Purdueuniversity,Indiana. Lindsay,B.(1982),\ConditionalScoreFunctions:SomeOptimalityResults,"Biometrika,69,503-512. Lindley,D.V.(1965),IntroductiontoProbabilityandStatisticsfromaBayesianViewpoint,CambridgeUniversityPress:Cambridge. McCullagh,P.,andTibshirani,R.(1990),\ASimpleMethodfortheAdjustmentofProleLikelihoods,"JournaloftheRoyalStatisticalSociety,Ser.B,52,325-344. Morgan,W.A.(1939),\ATestfortheSignicanceoftheDierenceBetweentheTwoVariancesinaSamplefromaNormalBivariatePopulation,"Biometrika,31,13-19. 95
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Mukerjee,R.,andGhosh,M.(1997),\SecondOrderProbabilityMatchingPriors,"Biometrika,84,970-975. Mukerjee,R.,andReid,N.(2001),\Second-OrderProbabilityMatchingPriorsforaParametricFunctionwithApplicationtoBayesianToleranceLimits,"Biometrika,88,587-592. Nicolaou,A.(1993),\BayesianIntervalswithGoodFrequencyBehaviorinthePresenceofNuisanceParameters,"JournaloftheRoyalStatisticalSociety,Ser.B,55,377-390. Peers,H.W.(1965),\CondencePropertiesofBayesianIntervalEstimates,"JournaloftheRoyalStatisticalSociety,Ser.B,30,535-544. Pitman,E.J.G.(1939),\ANoteonNormalCorrelation,"Biometrika,31,9-12. Rao,C.R.,andMukerjee,R.(1995),\OnPosteriorCredibleSetsbasedontheScoreStatistic,"StatisticaSinica,5,781-791. Roy,S.N.,andPottho,R.F.(1958),\CondenceBoundsonVectorAnaloguesofthe\RatioofMeans"andthe\RatioofVariances"forTwoCorrelatedNormalVariatesandSomeAssociatedTests,"TheAnnalsofMathematicalStatistics,29,829-841. Severini,T.A.(1991),\OntheRelationshipbetweenBayesianandNon-BayesianIntervalEstimates,"JournaloftheRoyalStatisticalSociety,Ser.B,53,611-618. Severini,T.A.,Mukerjee,R.,andGhosh,M.(2002),\OnanExactProbabilityMatchingPropertyofRight-InvariantPriors,"Biometrika,89,952-957. Staicu,A.(2007),\OnSomeAspectsofLikelihoodMethodswithApplicationsinBiostatistics,"UnpublishedPhDdissertation,UniversityofToronto,Toronto. Stein,C,(1985),\OntheCoverageProbabilityofCondenceSetsBasedonaPriorDistribution,"inSequentialMethodsinStatistics,BanachCenterPublications,16,Warsaw:PolishScienticPublishers,pp.485-514. Sun,D.,andBerger,J.(2006),\ObjectiveBayesianAnalysisfortheMultivariateNormalModel,"ISDSTechnicalReport,DukeUniversity.ToappearinBayesianStatistics8,eds.J.M.Bernardo,et.al.,Oxford,U.K.,OxfordUniversityPress. Tibshirani,R.(1989),\NoninformativePriorsforOneParameterofMany,"Biometrika,76,604-608. 96
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Yin,M.,andGhosh,M.(1997),\ANoteontheProbabilityDierenceBetweenMatchingPriorsBasedonPosteriorQuantilesandonInversionofConditionalLikelihoodRatioStatistics,"CalcuttaStatisticalAssociationBulletin,47,59-65. 97
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UpasanaSantrawasbornonMarch4,1977inKanpur,India.ShegraduatedfromSt.Mary'sConventHighSchool,Kanpurin1995.SheearnedherB.Sc.fromBanarasHinduUniversity,VaranasiandherM.Sc.fromIndianInstituteofTechnology,Kanpurin1998and2000,respectively,majoringinStatistics.UponarrivingtotheUnitedStateswithherhusband,SwadeshmukulSantra,sheworkedasaStatisticalConsultantintheStatisticsUnitofIFASattheUniversityofFlorida.SheearnedherM.S.inStatisticsin2003fromtheUniversityofFloridaandcontinuedforherPh.D.degreethereafter. 98
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