下载

0下载券

加入VIP
  • 专属下载特权
  • 现金文档折扣购买
  • VIP免费专区
  • 千万文档免费下载

上传资料

关闭

关闭

关闭

封号提示

内容

首页 CD

CD.pdf

CD

michaelleahcim
2013-10-20 0人阅读 举报 0 0 0 暂无简介

简介:本文档为《CDpdf》,可适用于高等教育领域

IMSLectureNotes–MonographSeriesComplexDatasetsandInverseProblems:Tomography,NetworksandBeyondVol()–c©InstituteofMathematicalStatistics,DOI:Confidencedistribution(CD)–distributionestimatorofaparameterKesarSingh,∗,MingeXie,†andWilliamEStrawderman,‡RutgersUniversityAbstract:Thenotionofconfidencedistribution(CD),anentirelyfrequentistconcept,isinessenceaNeymanianinterpretationofFisher’sFiducialdistributionItcontainsinformationrelatedtoeverykindoffrequentistinferenceInthisarticle,aCDisviewedasadistributionestimatorofaparameterThisleadsnaturallytoconsiderationoftheinformationcontainedinCD,comparisonofCDsandoptimalCDs,andconnectionoftheCDconcepttothe(profile)likelihoodfunctionAformaldevelopmentofamultiparameterCDisalsopresentedIntroductionandtheconceptWearehappytodedicatethisarticletothememoryofourcolleagueYehudaVardiHewassupportiveofoureffortstodevelopthisresearchareaandinparticularbroughthispaperwithColinMallows(MallowsandVardi)toourattentionduringthediscussionAconfidencedistribution(CD)isacompactexpressionoffrequentistinferencewhichcontainsinformationonjustabouteverykindofinferentialproblemTheconceptofaCDhasitsrootsinFisher’sfiducialdistribution,althoughitisapurelyfrequentistconceptwithapurelyfrequentistinterpretationSimplyspeaking,aCDofaunivariateparameterisadatadependentdistributionwhosesthquantileistheupperendofaslevelonesidedconfidenceintervalofθThisassertionclearlyentailsthat,forany<s<t<,theintervalformedbysthandtthquantilesofaCDisa(t−s)leveltwosidedconfidenceintervalThus,aCDisinfactNeymanianinterpretationofFisher’sfiducialdistribution(Neyman)TheconceptofCDhasappearedinanumberofresearcharticlesHowever,themodernstatisticalcommunityhaslargelyignoredthenotion,particularlyinapplicationsWesuspecttwoprobablecausesliebehindthis:(I)ThefirstisitshistoricconnectiontoFisher’sfiducialdistribution,whichislargelyconsideredas“Fisher’sbiggestblunder”(see,forinstance,Efron)(II)StatisticianshavenotseriouslylookedatthepossibleutilityofCDsinthecontextofmodernstatisticalpracticeAspointedoutbySchwederandHjort,therehasrecentlybeenarenewedinterestinthistopicSomerecentarticlesincludeEfron,,Fraser,,Lehmann,SchwederandHjort,,Singh,XieandStrawderman,,amongothersInparticular,recentarticlesemphasizetheNeymanianinterpretationoftheCDandpresentitasavaluablestatisticaltoolforinference∗ResearchpartiallysupportedbyNSFDMS†ResearchpartiallysupportedbyNSFSES‡ResearchpartiallysupportedbyNSAGrantGDepartmentofStatistics,HillCenter,BuschCampus,RutgersUniversity,Piscataway,NJ,USA,email:kesarstatrutgersedumxiestatrutgersedustarwstatrutgerseduurl:statrutgerseduAMSsubjectclassifications:primaryF,F,GsecondaryG,GKeywordsandphrases:confidencedistribution,frequentistinference,fiducialinference,optimality,bootstrap,datadepth,likelihoodfunction,pvaluefunctionConfidencedistributionsForexample,SchwederandHjortproposedreducedlikelihoodfunctionfromtheCDsforinference,andSingh,XieandStrawdermandevelopedattractivecomprehensiveapproachesthroughtheCDstocombininginformationfromindependentsourcesThefollowingquotationfromEfrononFisher’scontributionoftheFiducialdistributionseemsquiterelevantinthecontextofCDs:“buthereisasafepredictionforthestcentury:statisticianswillbeaskedtosolvebiggerandmorecomplicatedproblemsIbelievethereisagoodchancethatobjectiveBayesmethodswillbedevelopedforsuchproblems,andthatsomethinglikefiducialinferencewillplayanimportantroleinthisdevelopmentMaybeFisher’sbiggestblunderwillbecomeabighitinthestcentury!”Intheremainderofthissection,wegiveaformaldefinitionofaconfidencedistributionandtheassociatednotionofanasymptoticconfidencedistribution(aCD),provideasimplemethodofconstructingCDsaswellasseveralexamplesofCDsandaCDsInthefollowingformaldefinitionoftheCDs,nuisanceparametersaresuppressedfornotationalconvenienceItistakenfromSingh,XieandStrawderman,TheCDdefinitionisessentiallythesameasinSchwederandHjorttheydidnotdefinetheasymptoticCDhoweverDefinitionAfunctionHn(·)=Hn(Xn,·)onX×Θ→,iscalledaconfidencedistribution(CD)foraparameterθ,if(i)ForeachgivensamplesetXninthesamplesetspaceX,Hn(·)isacontinuouscumulativedistributionfunctionintheparameterspaceΘ(ii)Atthetrueparametervalueθ=θ,Hn(θ)=Hn(Xn,θ),asafunctionofthesamplesetXn,hasauniformdistributionU(,)ThefunctionHn(·)iscalledanasymptoticconfidencedistribution(aCD),ifrequirement(ii)aboveisreplacedby(ii)′:Atθ=θ,Hn(θ)W−→U(,),asn→∞,andthecontinuityrequirementonHn(·)isdroppedWecall,whenitexists,hn(θ)=H′n(θ)aCDdensityItisalsoknownasconfidencedensityintheliteratureItfollowsfromthedefinitionofCDthatifθ<θ,Hn(θ)sto≤−Hn(θ),andifθ>θ,−Hn(θ)sto≤Hn(θ)Here,sto≤isastochasticcomparisonbetweentworandomvariablesie,fortworandomvariableYandY,Ysto≤Y,ifP(Y≤t)≥P(Y≤t)foralltThusaCDworks,inasense,likeacompassneedleItpointstowardsθ,whenplacedatθ�=θ,byassigningmoremassstochasticallytothatside(leftorright)ofθthatcontainsθWhenplacedatθitself,Hn(θ)=Hn(θ)hastheuniformU,distributionandthusitisnoninformativeindirectionTheinterpretationofaCDasadistributionestimatorisasfollowsThepurposeofanalyzingsampledataistogatherknowledgeaboutthepopulationfromwhichthesamplecameTheunknownθisacharacteristicofthepopulationThoughuseful,theknowledgeacquiredfromthedataanalysisisimperfectinthesensethatthereisstilla,usuallyknown,degreeofuncertaintyremainingStatisticianscanpresenttheacquiredknowledgeonθ,withtheleftoveruncertainty,intheformofaprobabilitydistributionThisappropriatelycalibrateddistribution,thatreflectsstatisticians’confidenceregardingwhereθlives,isaCDThus,aCDisanexpressionofinference(aninferentialoutput)andnotadistributiononθWhatisreallyfascinatingisthataCDisloadedwithawealthofinformationaboutθ(asitisdetailedlater),asisaposteriordistributioninBayesianinferenceBeforewegivesomeillustrativeexamples,letusdescribeageneralsubstitutionschemefortheconstructionofCDs,thatavoidsinversionoffunctionsSee,alsoSchwederandHjortAlthoughthisschemedoesnotcoverallpossiblewaysofconstructingCDs(see,forexample,Section),itcoversawiderangeofexamplesKSingh,MXieandWEStrawdermaninvolvingpivotalstatisticsConsiderastatisticalfunctionψ(Xn,θ)whichinvolvesthedatasetXnandtheparameterofinterestθBesidesθ,thefunctionψmaycontainsomeknownparameters(whichshouldbetreatedasconstants)butitshouldnothaveanyotherunknownparameterOnψ,weimposethefollowingcondition:•ForanygivenXn,ψ(Xn,θ)iscontinuousandmonotonicasafunctionofθSupposefurtherthatGn,thetruecdfofψ(Xn,θ),doesnotinvolveanyunknownparameteranditisanalyticallytractableInsuchacase,ψ(Xn,θ)isgenerallyknownasapivotThenonehasthefollowingexactCDforθ(providedGn(·)iscontinuous):Hn(x)={Gn(ψ(Xn,x)),ifψisincreasinginθ−Gn(ψ(Xn,x)),ifψisdecreasinginθInmostcases,Hn(x)istypicallyacontinuouscdfforfixedXnand,asafunctionofXn,Hn(θ)followsaU,distributionThus,HnisaCDbydefinitionNotethesubstitutionofθbyxIncasethesamplingdistributionGnisunavailable,includingthecaseinwhichGndependsonunknownnuisanceparameters,onecanturntoanapproximateorestimatedsamplingdistributionGˆnThiscouldbethelimitofGn,anestimateofthelimitoranestimatebasedonbootstraporsomeothermethodUtilizingGˆn,onedefinesHn(x)={Gˆn(ψ(Xn,x)),ifψisincreasinginθ,−Gˆn(ψ(Xn,x)),ifψisdecreasinginθInmostcases,Hn(θ)L−→U,andisthusanasymptoticCDTheaboveconstructionresemblesBeran’sconstructionofprepivot(seeBeran,page),whichwasdefinedatθ=θ(thetruevalueofθ)Beran’sgoalwastoachievesecondorderaccuracyingeneralviadoublebootstrapWenowpresentsomeillustrativeexamplesofCDsExample(Normalmeanandvariance)ThemostbasiccaseisthatofsamplingfromanormaldistributionwithparametersµandσConsiderfirstaCDforthemeanwhenthevarianceisunknownHerethestandardpivotisψ(Xn,µ)=(X¯n−µ)(sn√n),whichhasthestudenttdistributionwith(n−)dfUsingtheabovesubstitution,theCDforµisHn(x)=−P(Tn−≤X¯−xsn√n)=P(Tn−≤x−X¯sn√n),whereTn−isarandomvariablethathastheStudent’stn−distributionForσ,theusualpivotisψ(Xn,σ)=(n−)snσBythesubstitutionmethod,theCDforσis()Hn(x)=P(χn−≥(n−)snx),x≥whereχn−isarandomvariablethathastheChisquaredistributionwithn−degreesoffreedomConfidencedistributionsExample(Bivariatenormalcorrelation)Forabivariatenormalpopulation,letρdenotethecorrelationcoefficientTheasymptoticpivotusedinthisexampleisFisher’sZ,ψ(Xn,ρ)=log((r)(−r))−log(ρ)(−ρ)whereristhesamplecorrelationItslimitingdistributionisN(,n−),withafastrateofconvergenceSotheresultingasymptoticCDisHn(x)=−Φ(√n−(logr−r−logx−x)),−≤x≤Example(Nonparametricbootstrap)Turningtononparametricexamplesbasedonbootstrap,letθˆbeanestimatorofθ,suchthatthelimitingdistributionofθˆ,properlynormalized,issymmetricUsingsymmetry,ifthesamplingdistributionofθˆ−θisestimatedbythebootstrapdistributionofθˆ−θˆB,thenanasymptoticCDisgivenbyHn(x)=−PB(θˆ−θˆB≤θˆ−x)=PB(θˆB≤x)Here,θˆBisθˆcomputedonabootstrapsampleTheresultingasymptoticCDistherawbootstrapdistributionofθˆIfthedistributionofθˆ−θisestimatedbythebootstrapdistributionofθˆB−θˆ,whichiswhatbootstrappersusuallydo,thecorrespondingasymptoticCDisHn(x)=−PB(θˆB−θˆ≤θˆ−x)=PB(θˆB≥θˆ−x)Example(Bootstraptmethod)Bythebootstraptmethod,thedistributionofasymptoticpivot(θˆ−θ)ŜE(θˆ)isestimatedbythebootstrapdistributionof(θˆB−θˆ)ŜEB(θˆB)HereŜEB(θˆB)istheestimatedstandarderrorofθˆB,basedonthebootstrapsampleSuchanapproximationhassocalled,secondorderaccuracy(seeSingh,BabuandSingh,)TheresultingasymptoticCDwouldbeHn(x)=−PB(θˆB−θˆŜEB(θˆB)≤θˆ−xŜE(θˆ))SuchaCD,atx=θ,typicallyconvergestoU,,inlaw,atarapidpaceExample(BootstraprdorderaccurateaCD)Hallcameupwiththefollowingincreasingfunctionofthetstatistics,whichdoesnothavethe√nterminitsEdgeworthexpansion:ψ(Xn,µ)=tλˆ√n(t)nλˆtHeret=√n(X¯−µ)sn,λ=µσ,λˆisasampleestimateofλandtheassumptionofpopulationnormalityisdroppedUndermildconditionsonthepopulationdistribution,thebootstrapapproximationtothedistributionofthisfunctionoft,isthirdordercorrectLetGˆBbethecdfofthebootstrapapproximationThen,usingthesubstitution,asecondordercorrectCDforµisgivenbyHn(x)=−GˆB(ψ(Xn,x))OnealsohasCDsthatdonotinvolvepivotalstatisticsAparticularclassofsuchCDsareconstructedfromlikelihoodfunctionsWewillhavesomedetaileddiscussionsontheconnectionsofCDsandlikelihoodfunctionsinSectionForeachgivensampleXn,Hn(·)isacumulativedistributionfunctionWecanconstructarandomvariableξsuchthatξhasthedistributionHnForconvenienceofpresentations,wecallξaCDrandomvariableKSingh,MXieandWEStrawdermanDefinitionWecallξ=ξHnaCDrandomvariableassociatedwithaCDHn,iftheconditionaldistributionofξgiventhedataXnisHnAsanexample,letUbeaU(,)randomvariableindependentofXn,thenξ=H−n(U)isaCDrandomvariableLetusnotethatξmaybeviewedasaCDrandomizedestimatorofθAsanestimator,ξismedianunbiased,ie,Pθ(ξ≤θ)=Eθ{Hn(θ)}=However,ξisnotalwaysmeanunbiasedForexample,theCDrandomvariableξassociatedwith()inExampleismeanbiasedasanestimatorofσWeclosethissectionwithaequivarianceresultonCDs,whichmaybehelpfulintheconstructionofaCDforafunctionofθForexample,toderiveaCDforσfromthatofσgiveninExampleTheequivarianceissharedbyEfron’sbootstrapdistributionofanestimator,whichisofcourseaCD(Exampleabove)underconditionsPropositionLetHnbeaCDforθandξbeanassociatedCDrandomvariableThen,theconditionaldistributionfunctionofg(ξ),forgivenXn,isaCDofg(θ),ifgismonotonicWhenthemonotonicityislimitedtoaneighborhoodofθonly,thentheconditionaldistributionofg(ξ),forgivenXn,yieldsanasymptoticCDatθ=θ,provided,forall�>,Hn(θ�)−Hn(θ−�)p−→ProofTheproofofthefirstclaimisstraightforwardForthesecondclaim,wenotethat,ifg(·)isincreasingwithin(θ−�,θ�),P(g(ξ)≤g(θ)|x)=P({ξ≤θ}∩{θ−�≤ξ≤θ�}|x)op()=Hn(θ)op()Onearguessimilarlyfordecreasingg(·)TherestofthepaperisarrangedasfollowsSectionisdevotedtocomparingCDsforthesameparameterandrelatedissuesInSection,weexplore,fromthefrequentistviewpoint,inferentialinformationcontainedwithinaCDInSection,weestablishthatthenormalizedprofilelikelihoodfunctionisanaCDLastly,SectionisanattempttoformallydefineanddevelopthenotionofjointCDforaparametervectorPartsofSectionsandarecloselyrelatedtotherecentpaperofSchwederandHjort,andalsotoSingh,XieandStrawderman,SchwederandHjortpresentessentiallythesamedefinitionoftheCDandalsocompareCDsaswedointhispaper(SeeDefinition)TheyalsodevelopthenotionofanoptimalCDwhichisquiteclosetothatpresentedhereandinSingh,XieandStrawdermanOurdevelopmentisbasedonthetheoryofUMPUtestsanddiffersslightlyfromtheirsThematerialsonpvaluesinSectionisalsocloselyrelatedto,butsomewhatmoregeneralthan,thatofFraserComparisonofCDsandanotionofoptimalCDTheprecisionofaCDcanbemeasuredintermsofhowlittleprobabilitymassaCDwastesonsetsthatdonotincludeθThissuggeststhat,for�>,oneshouldcomparethequantitiesH(θ−�)withH(θ−�)andalso−H(θ�)and−H(θ�)Ineachcase,asmallervalueispreferredHereHandHareanytwoCDsforthecommonparameterθ,basedonthesamesampleofsizenDefinitionGiventwoCDsHandHforθ,wesayHismoreprecisethanH,atθ=θ,ifforall�>,H(θ−�)sto≤H(θ−�)and−H(θ�)sto≤−H(θ�)whenθistheprevailingvalueofθConfidencedistributionsAnessentiallyequivalentdefinitionisalsousedinSingh,XieandStrawdermanandSchwederandHjortThefollowingpropositionfollowsimmediatelyfromthedefinitionPropositionIfHismoreprecisethanHandtheybotharestrictlyincreasing,thenforalltin,,H−(t)−θsto≤H−(t)−θandH−(t)−θ−sto≤H−(t)−θ−Thus,()∣∣H−(t)−θ∣∣sto≤∣∣H−(t)−θ∣∣Thestatement()yieldsacomparisonofconfidenceintervalsbasedonHandHIngeneral,anendpointofaconfidenceintervalbasedonHisclosertoθthanthatbasedonHInsomesense,itimpliesthattheHbasedconfidenceintervalsaremorecompactAlso,theCDmedianofamorepreciseCDisstochasticallyclosertoθLetφ(x,θ)bealossfunctionsuchthatφ(·,·)isnondecreasingforx≥θandnonincreasingforx≤θWenowconnecttheabovedefinedCDcomparisontothefollowingconceptoftheφdispersionofaCDDefinitionForaCDH(x)ofaparameterθ,theφdispersionofH(x)isdefinedasdφ(θ,H)=Eθ∫φ(x,θ)dH(x)Inthespecialcaseofsquareerrorloss,dsq(θ,H)=Eθ∫(x−θ)dH(x)Ingeneral,wehavethefollowing:TheoremIfHismoreprecisethanHatθ=θ,intermsofDefinition,then()dφ(θ,H)≤dφ(θ,H)Infact,theabovetheoremholdsunderasetofweakerconditions:Forany�>,()E{H(θ−�)}≤E{H(θ−�)}andE{H(θ�)}≥E{H(θ�)}ProofTheclaimin()isequivalentto()E{φ(ξ,θ)}≤E{φ(ξ,θ)},whereξandξareCDrandomvariablesassociatedwithHandH(see,Definition),respectivelyFrom(),viaconditioningonXn,itfollowsthat(ξ−θ)sto≤(ξ−θ)and(ξ−θ)−sto≤(ξ−θ)−Duetothemonotonicityofφ(·,θ),wehaveφ(ξ,θ)I(ξ≥θ)sto≤φ(ξ,θ)I(ξ≥θ)andφ(ξ,θ)I(ξ<θ)sto≤φ(ξ,θ)I(ξ<θ)Theaboveinequalitiesleadto()immediatelyKSingh,MXieandWEStrawdermanSupposenowthatthereisafamilyofUniformlyMostPowerfulUnbiasedtestsfortestingK:θ≤θversusK:θ>θ,foreveryθTheunderlyingfamilyofdistributionsmayhavenuisanceparameter(s)Letthecorrespondingpvalue(theinfofαatwhichKcanberejected)p(θ)=p(Xn,θ)bestrictlyincreasingandcontinuousasafunctionofθItisfurtherassumedthat−p(θ)isthepvalueofanUMPUtestfortestingK:θ≥θvsK:θ<θLetthedistributionofp(θ)underθbeU,andlettherangeofp(·)be,DefinethecorrespondingCD,H∗(x)=p(Xn,x)WehavethefollowingresultTheoremTheCDH∗definedaboveismoreprecisethananyotherCDfortheparameterθ,atallθProofLetθ=θbethetruevalueNotethatPθ(H∗(θ−�)<α)isthepower(atθ=θ)oftheUMPUtestwhenKisθ≤θ−�andKisθ>θ−�GivenanyotherCDH,onehasthefollowingunbiasedtestfortestingthesamehypotheses:RejectKiffH(θ−�)<αTherefore,Pθ(H∗(θ−�)<α)≥Pθ(H(θ−�)<α)forallα∈,Usingthefunction−p(·),onesimilarlyarguesforPθ(−H∗(θ�)<α)≥Pθ(−H(θ�)<α)Thus,H∗ismostpreciseItshouldbementionedthatthepropertyofCDsasexhibitedinTheoremdependoncorrespondingoptimalitypropertiesofhypothesistestsThebasicideasbehindthissegmentcouldbetracedtothediscussionsofconfidenceintervalsinLehmannRemarkIftheunderlyingparametricfamilyhasthesocalledMLR(monotonelikelihoodratio)property,thereexistsanUMPtestforonesidedhypotheseswhosepvalueismonotonicExampleInthetestingproblemofnormalmeans,theZtestisUMPU(actuallyUMP),fortheonesidedhypotheseswhenσisknownThettestisUMPUfortheones

用户评价(0)

关闭

新课改视野下建构高中语文教学实验成果报告(32KB)

抱歉,积分不足下载失败,请稍后再试!

提示

试读已结束,如需要继续阅读或者下载,敬请购买!

评分:

/19

VIP

在线
客服

免费
邮箱

爱问共享资料服务号

扫描关注领取更多福利