ON A PROPERTY OF PROBABILITY MATCHING PRIORS MATCH

ONAPROPERTYOFPROBABILITYMATCHINGPRIORS:MATCHINGTHEALTERNATIVECOVERAGEPROBABILITIESRahulMukerjeeIndianInstituteofManagementPostBoxNo16757,Calcutta700027,[e-mail]csUniversityofTorontoToronto,[e-mail]ferencebasedoncondencesets,boththetruecoverageandtheprobabilityforacondencesettoincludeanalternativevalueoftheparameterofinterestareimportant.Thusifprobabilitymatchingpriorsalsomatchsuchalternativecoverageprobabilitiesthereisperhapsastrongerjusticationforcallingthemnoninformative.Con-sideringcontiguousalternatives,weobtaintherelevantnecessaryandsucientconditions.Inparticular,withndenotingthesamplesize,itisseenthatprobabilitymatchingpriorsuptoo(n1=2)alsomatchthealternativecoverageprobabilitiesuptothatorderwhilethisisnotnecessarilythecasewithprobabilitymatchingpriorsuptoo(n1).Somekeywords:Contiguousalternative;Firstorder;Jereys’prior;Noninformativeprior;parametricorthogonality;Secondorder.1.Introduction1Inrecentyears,therehasbeenconsiderableinterestinthecharacterisationofpriorsensuring,undersuitableregularityconditions,approximatefrequentistvalidityofposteriorcrediblesets;seeReid(1995)andKass&Wasserman(1996)forreviews.Inparticular,priors()forwhichtherelationPf1(1)1(;X)g=1+o(nr=2);(1:1)holdsforr=1or2andforeach(01),havereceivedmuchattention.Herenisthesamplesize,=(1;:::;p)0isanunknownparametervector,1istheone-dimensionalparameterofinterest,Pisthefrequentistprobabilitymeasureunder,and(1)1(;X)isthe(1)thposteriorquantileof1giventhedataX.Priorssatisfying(1.1)forr=1or2arecalledrst-orsecond-orderprobabilitymatchingpriorsrespectively.Forp=1,i.e.intheabsenceofnuisanceparameters,Welch&Peers(1963)characterisedJereys’priorasrst-orderprobabilitymatchingandexploredmodelconditionsunderwhichitisalsosecond-orderprobabilitymatching.Thecorrespondingproblemsforp2havebeeninvestigatedby,amongothers,Peers(1965),Tibshirani(1989),Nicolaou(1993)andMukerjee&Dey(1993).AsnotedbyTibshirani(1989),studiesofthiskindhelpingettingaccuratefrequentistcondencesetsanddeningnoninformativepriors.Fromthefrequentistpointofview,however,theprobabilityforacondencesettoincludeanalternativevalueoftheinterestparameterisasimportantasthatoftruecoverage(Lehmann,1986,Ch.3).Suchanalternativecoverageprobabilityindicateshowselectiveacondencesetis,anditscomplementislinkedwiththepoweroftheassociatedtest.Weconsiderinthispaperhowfarapriorsatisfying(1.1)alsomatchesPf1+(I11=n)1=2(1)1(;X)gwiththecorrespondingposteriorprobability,uptothesame2orderofapproximationandforeachand,whereI11isthe(1;1)elementoftheinverseoftheperobservationexpectedinformationmatrixatandthescalarisfreefromn,andX.Ifapriorsatisfying(1.1)alsomatchesthealternativecoverageprobabilitiesintheabovesense,thenthereisastrongerjusticationforcallingitnoninformativeinsofarasagreementwithafrequentistisconcerned.Afterpresentingthepreliminariesinx2,weobtainnecessaryandsucientconditionsformatchingtrueandalternativecoverageprobabilities,uptotherstandsecondordersofapproximation,inx3.Itisseenthatrst-orderprobabilitymatchingpriorsalsomatchalternativecoverageprobabilitiesuptothatorderwhilethatisnotnecessarilythecaseatthesecondorderofapproximation.Finally,someillustrativeexamples,includingonewherethepresentstudyhelpsinnarrowingdowntheclassofsecond-orderprobabilitymatchingpriors,arepresentedinx4.2.NotationandpreliminaryresultsLetfXig,i1,beasequenceofindependentandidenticallydistributedpossiblyvector-valuedrandomvariableswithcommondensityf(x;),wheretheparametervector=(1;:::;p)0belongstoRporsomeopensubsetthereofand1istheparameterofinterest.AlongthelinesofMukerjee&Dey(1993),weworkessentiallyundertheassumptionsofJohnson(1970)andalsoneedtheEdgeworthassumptionsofBickel&Ghosh(1990,p.1078).Allformalexpansionsfortheposterior,asusedhere,arevalidforsamplepointsinasetSwithPprobability1+o(n1)uniformlyovercompactsetsof.ThesetSmaybedenedfollowingBickel&Ghosh(1990,Section2).3Let^=(^1;:::;^p)0bethemaximumlikelihoodestimatorofbasedonX=(X1;:::;Xn)0,‘()=n1nPi=1logf(Xi;)and,withDj@=@j,letajr=fDjDr‘()g=^;ajrs=fDjDrDs‘()g=^;cjr=ajr;Vj=Djlogf(X1;);Vjr=DjDrlogf(X1;);Vjrs=DjDrDslogf(X1;);Ijr=E(VjVr);Lj;r;s=E(VjVrVs);Lj;rs=E(VjVrs);Ljrs=E(Vjrs):ThematrixC=(cjr)ispositivedeniteoverS.LetC1=(cjr)andkjr=cjr(cj1cr1=c11).Similarly,letI=(Ijr)betheperobservationFisherinformationmatrixat,I1=(Ijr),jr=Ij1Ir1=I11andjr=Ijrjr.ThequantitiesIjr,Ijr,jr,jr,Lj;r;s,etc.areallfunctionsof.Lethaveapriordensity()whichispositiveandthricecontinuouslydierentiableforall.Wedenej()=Dj(),jr()=DjDr(),^=(^),^j=j(^),^jr=jr(^).Similarly,let()=(I11)1=2,j()=Dj(),jr()=DjDr(),^=(^),^j=j(^),^jr=jr(^).Asindicatedinx1,thequantity1+(I11=n)1=2,wheredoesnotinvolven,orX,willplayacrucialroleinthepresentwork.Tofacilitatethepresentation,weconsiderarelatedquantity=(c11)1=2[n1=2f1+(I11=n)1=2^1g^];(2:1)and,recallingthedenitionof(),notethat=(c11)1=2fh1+(n1=2