J. K. Ghosh's contribution to statistics A brief outline

  • 格式:pdf
  • 大小:260.78 KB
  • 文档页数:18

arXiv:0805.3066v1 [math.ST] 20 May 2008IMSCollectionsPushingtheLimitsofContemporaryStatistics:ContributionsinHonorofJayantaK.GhoshVol.3(2008)1–18c󰀁InstituteofMathematicalStatistics,2008

DOI:10.1214/074921708000000011

J.K.Ghosh’scontributiontostatistics:Abriefoutline

BertrandClarke1andSubhashisGhosal2UniversityofBritishColumbiaandNorthCarolinaStateUniversityAbstract:ProfessorJayantaKumarGhoshhascontributedmassivelytovar-iousareasofStatisticsoverthelastfivedecades.Here,wesurveysomeofhismostimportantcontributions.Inroughlychronologicalorder,wediscusshismajorresultsintheareasofsequentialanalysis,foundations,asymptotics,andBayesianinference.Itisseenthatheprogressedfromthinkingaboutdatapoints,tothinkingaboutdatasummarization,tothelimitingcasesofdatasummarizationinastheyrelatetoparameterestimation,andthentomoregeneralaspectsofmodelingincludingpriorandmodelselection.

Contents1Introduction...................................22Sequentialanalysis...............................23Foundationsofstatistics............................34Asymptotics...................................44.1Bahadur–Ghosh–Kieferrepresentation.................54.2Edgeworthexpansions..........................54.3Secondorderefficiency..........................64.4Bartlettcorrection............................74.5Comparisonofthelikelihoodratio,Wald’sandRao’sstatistics...74.6Bahadur–Cochrandeficiency.......................84.7Neyman–Scottproblemandsemiparametricinference........85Bayesianinference...............................85.1Matchingandotherobjectivepriors..................95.2Limitsofposteriordistributions.....................105.3Bayesiannonparametrics.........................125.4ModelselectionandBayesianhypothesistesting...........136Concludingremarks..............................14References......................................152B.ClarkeandS.Ghosal1.IntroductionProfessorJayantaKumarGhosh,orJ.K.Ghosh,asheiscommonlyknown,hasbeenaprominentcontributortothedisciplineofstatisticsforfivedecades.Thespectrumofhiscontributionsencompassessequentialanalysis,thefounda-tionsofstatistics,finitepopulations,Edgeworthexpansions,secondorderefficiency,Bartlettcorrections,noninformative,andespeciallymatching,priors,semiparamet-ricinference,posteriorlimittheorems,Bayesiannonparametrics,modelselectionBayesianhypothesistestingandhighdimensionaldataanalysis,aswellassomeappliedworkinreliabilitytheory,statisticalqualitycontrol,modelinghydrocarbondiscoveries,geologicalmappingandDNAfingerprinting.Byitself,coveringsuchdiversetopicsindepthisamajorcareerachievement.Hehasauthoredover130publicationsincludingthreemonographsandseveraleditedvolumes.Hisbooks,oneentitledHigherOrderAsymptoticsandpublishedasanIMSmonographandanotherentitledBayesianNonparametrics,co-authoredbyR.V.RamamoorthiandpublishedbySpringer-Verlag,continuetoholdrespectedpositionsforresearchersintheseareas.Hisrecentlypublishedthirdbook[34]isafinegraduatetextonBayesianinference.Inaddition,hisservicetotheprofession,especiallyastheedi-torofSankhy¯a,hasbeeninvaluable.Thevarietyofhisworknotwithstanding,asymptoticshasbeencentraltohisthinkingacrossawiderangeofproblems.Accordingly,inwhatfollows,weoutlinesomeofhiswork,inroughlychronologicalorder,focussingonthosecontributionswhichareintimatelyconnectedtoasymptotics.Inthecourseofreviewinghiswork,wetrytocharacterizetheprogressionofthinkingthatnaturallyconnectsthetopicsthatJ.K.Ghoshhasdonesomuchtodevelop.

2.SequentialanalysisJ.K.GhoshstartedhisresearchcareerinSequentialAnalysisintheearlysixtiesasaGraduateStudentintheDepartmentofStatisticsatCalcuttaUniversity.Waldhadrecentlyintroducedhissequentialprobabilityratiotest(SPRT),butitspropertieswerenotwellunderstoodinthecompositecase.ThiswasthefirsttopictowhichGhoshturnedhisattention.Throughhiswork,manyofthepropertiesofSPRTandrelatedprocedureswereestablishedandbetterunderstood.Forinstance,inthetestingcontext,doubleminimaxityessentiallymeanssimultaneousminimizationofaveragetypeIandIIerrorprobabilities.Inhisfirstpublishedwork[26],Ghosh

clarifiedaresultofWaldonthedoubleminimaxityoftheSPRTfornormaltwo-sidedalternativehypothesis(withunknownscale)separatedfromthenullbyδ.Itiswell-knownthatthepowerfunctionismonotonicinmanycommonfamiliesforfixedsamplesizes.Ghoshestablishedananalogofthisresultin[27],namely

thattheoperatingcharacteristicfunctionofthe(generalized)SPRTcontinuestobemonotonic.Alsointhesequentialcontext,[28]consideredtheadmissibilityofsequentialtestsbasedonasimpleidentitywhichlaterbecameknownastheGhosh–Prattidentity.GhoshcomparedtheSPRTnotjustwiththeclassofalltestswithfiniteexpectedsamplesizebutalsowithinotherclasses,forinstance,theclasswhichrequiresatleastoneobservationorwhichrequiresnomorethanapredeterminednumberofobservationstoreachaconclusion.Followingthis,GhoshcontinuedtoelucidatemorepropertiesoftheSPRT,anditsvariants,whichcouldbeseenasanalogsofthecorrespondingpropertiesNeyman–PearsonorBayestestsforfixedsamplesize.In[29],heprovedthatfor