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Probability Matching Priors for the Bivariate Normal Distribution

Permanent Link: http://ufdc.ufl.edu/UFE0021864/00001

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Title: Probability Matching Priors for the Bivariate Normal Distribution
Physical Description: 1 online resource (98 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

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Subjects / Keywords: adjusted, bivariate, conditional, distribution, first, hpd, likelihood, matching, order, probability, quantile, second
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: In practice, most Bayesian analyses are performed with so called 'non-informative' priors. This is especially so when there is little or no prior information, and yet the Bayesian technique can lead to solutions satisfactory from both the Bayesian and the frequentist perspectives. The study of probability matching priors ensuring, upto the desired order of asymptotics, the approximate frequentist validity of posterior credible sets has received significant attention in recent years. In this dissertation we develop some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, the ratio of one of the conditional variances to the marginal variance of the other variable, the correlation coefficient and the ratio of the standard deviations. The criterion used is the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. Various matching criteria, namely, quantile matching, matching of distribution functions, highest posterior density matching, and matching via inversion of test statistics are used. One particular prior is found which meets all the matching criteria individually for the regression coefficient, the generalized variance and the ratio of one of the conditional variances to the marginal variance of the other variable. For the correlation coefficient though, each matching criterion leads to a different prior. There however, does not exist a prior that satisfies the matching via distribution functions criterion in this case. Finally, a general class of priors have been obtained for inference about the ratio of standard deviations. The propriety of the resultant posteriors is proved in each case under mild conditions and simulation results suggest that the approximations are valid even for moderate sample sizes. Further, several likelihood based methods have been considered for the correlation coefficient. One common feature of all these modified likelihoods is that they are all dependent on the data only through the sample correlation coefficient r.
General Note: In the series University of Florida Digital Collections.
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Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Ghosh, Malay.
Local: Co-adviser: Mukherjee, Bhramar.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2008-11-30

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Permanent Link: http://ufdc.ufl.edu/UFE0021864/00001

Material Information

Title: Probability Matching Priors for the Bivariate Normal Distribution
Physical Description: 1 online resource (98 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: adjusted, bivariate, conditional, distribution, first, hpd, likelihood, matching, order, probability, quantile, second
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In practice, most Bayesian analyses are performed with so called 'non-informative' priors. This is especially so when there is little or no prior information, and yet the Bayesian technique can lead to solutions satisfactory from both the Bayesian and the frequentist perspectives. The study of probability matching priors ensuring, upto the desired order of asymptotics, the approximate frequentist validity of posterior credible sets has received significant attention in recent years. In this dissertation we develop some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, the ratio of one of the conditional variances to the marginal variance of the other variable, the correlation coefficient and the ratio of the standard deviations. The criterion used is the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. Various matching criteria, namely, quantile matching, matching of distribution functions, highest posterior density matching, and matching via inversion of test statistics are used. One particular prior is found which meets all the matching criteria individually for the regression coefficient, the generalized variance and the ratio of one of the conditional variances to the marginal variance of the other variable. For the correlation coefficient though, each matching criterion leads to a different prior. There however, does not exist a prior that satisfies the matching via distribution functions criterion in this case. Finally, a general class of priors have been obtained for inference about the ratio of standard deviations. The propriety of the resultant posteriors is proved in each case under mild conditions and simulation results suggest that the approximations are valid even for moderate sample sizes. Further, several likelihood based methods have been considered for the correlation coefficient. One common feature of all these modified likelihoods is that they are all dependent on the data only through the sample correlation coefficient r.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Ghosh, Malay.
Local: Co-adviser: Mukherjee, Bhramar.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2008-11-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0021864:00001


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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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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