Iterated smoothed bootstrap confidence intervals f

arXiv:math/0504516v1[math.ST]25Apr2005TheAnnalsofStatistics2005,Vol.33,No.1,437–462DOI:10.1214/009053604000000878cInstituteofMathematicalStatistics,2005ITERATEDSMOOTHEDBOOTSTRAPCONFIDENCEINTERVALSFORPOPULATIONQUANTILESByYvonneH.S.HoandStephenM.S.Lee1TheUniversityofHongKongThispaperinvestigatestheeffectsofsmoothedbootstrapitera-tionsoncoverageprobabilitiesofsmoothedbootstrapandbootstrap-tconfidenceintervalsforpopulationquantiles,andestablishestheopti-malkernelbandwidthsatvariousstagesofthesmoothingprocedures.Theconventionalsmoothedbootstrapandbootstrap-tmethodshavebeenknowntoyieldone-sidedcoverageerrorsofordersO(n−1/2)ando(n−2/3),respectively,forintervalsbasedonthesamplequantileofarandomsampleofsizen.WesharpenthelatterresulttoO(n−5/6)withproperchoicesofbandwidthsatthebootstrappingandStu-dentizationsteps.Weshowfurtherthatcalibrationofthenominalcoveragelevelbymeansoftheiteratedbootstrapsucceedsinreduc-ingthecoverageerrorofthesmoothedbootstrappercentileintervaltotheorderO(n−2/3)andthatofthesmoothedbootstrap-tintervaltoO(n−58/57),providedthatbandwidthsareselectedofappropriateorders.Simulationresultsconfirmourasymptoticfindings,suggest-ingthattheiteratedsmoothedbootstrap-tmethodyieldsthemostaccuratecoverage.Ontheotherhand,theiteratedsmoothedboot-strappercentilemethodintervalhastheadvantageofbeingshorterandmorestablethanthebootstrap-tintervals.1.Introduction.ItisgenerallyknownthatunderBhattacharyaandGhosh’s(1978)smoothfunctionmodel,thebootstrappercentilemethodconfidenceintervalissubjecttoaone-sidedcoverageerroroforderO(n−1/2),renderingitindistinguishablefromtheclassicalnormalapproximationmethod.Hall(1986)showsthatStudentizationcanbeemployedtoreducethiserrortoO(n−1).Theiteratedbootstrapoffersanalternativetoerrorcorrectionbycalibratingthenominalcoverageleveliteratively;seeBeran(1987).HallandReceivedApril2003;revisedMarch2004.1SupportedbyagrantfromtheResearchGrantsCounciloftheHongKongSpecialAdministrativeRegion,China(ProjectHKU7131/00P).AMS2000subjectclassifications.Primary62G15;secondary62F40,62G30.Keywordsandphrases.Bandwidth,bootstrap-t,iteratedbootstrap,kernel,quantile,smoothedbootstrap,Studentizedsamplequantile.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofStatistics,2005,Vol.33,No.1,437–462.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12Y.H.S.HOANDS.M.S.LEEMartin(1988)showthateachsuchiterationreducestheone-sidedcoverageerrorbyanorderofO(n−1/2)successively.Ontheotherhand,smoothingthebootstrap,whichamountstodrawingbootstrapsamplesfromakernel-smoothedempiricaldistribution,insteadofsamplingwithreplacementfromtherawdata,doesnotaffectthecoverageaccuracyofbootstrapintervalsingeneral.PolanskyandSchucany(1997)proposesmoothedbootstrapstrate-giestoyieldintervalsofO(n−1)coverageerror.Theirmethods,however,requiresophisticatedtuningofthesmoothingbandwidths,renderingtheimprovementlessstablethanthatresultingfromStudentizationortheiter-atedbootstrap.Incontrasttoproblemsinthecontextofsmoothfunctionmodels,con-ventionalbootstrapconfidenceintervalsfortheqthpopulationquantile,forafixedq∈(0,1),havenotablypoorcoverages;seeHallandMartin(1989).HerethepercentilemethodgivescoverageerroroforderO(n−1/2),whichcannotbeimproveduponbynominalcoveragecalibrationusingtheiteratedbootstrap.Indeed,moregenerally,thisO(n−1/2)coverageerrorisinherentinanyconfidenceintervalprocedurebaseddirectlyonorderstatisticsasaconsequenceoftheirbinomial-typediscreteness.See,forexample,DeAn-gelis,HallandYoung(1993)foramoredetailedaccountoftheabovephe-nomenon.Ontheotherhand,eitherthesmoothedbootstraporStudentiza-tionextendsconsiderablythedomainfromwhichwederivetheconfidencelimits,andmaythereforebeabletomakeasymptoticimprovementovertheconventionalbootstrappercentilemethod.Inthecontextofestimatingthevarianceσ2nofthesampleqthquan-tile,Hall,DiCiccioandRomano(1989)showthatsufficientlyhigh-ordersmoothingofthebootstrapsucceedsinreducingtherelativeerroroftheunsmoothedbootstrapestimatefromO(n−1/4)toO(n−1/2+ǫ)forarbitrar-ilysmallǫ0.FalkandJanas(1992)showthatsmoothingthebootstrappercentilemethodreturnsthesameorder,O(n−1/2),ofcoverageerrorasintheunsmoothedcase.Studentizationofthesamplequantileinvolvesestima-tionofσ2n,whichadmitsanexpansionn−1q(1−q)f(F−1(q))−2+O(n−3/2)underproperregularityconditions,wheref=F′andFdenotesthedistri-butionfunctionunderlyingthegivenrandomsample.Inpracticeσ2nhastobeestimatedfromthesampleby,forexample,bootstrappingorexplicites-timationoftheleadingterminitsexpansionabove.HallandMartin(1991)showthatconfidenceintervalsbasedonnormalapproximationofthesam-plequantileStudentizedbytheunsmoothedbootstrapvarianceestimateyieldcoverageerroroforderO(n−1/2).FalkandJanas(1992)showthatasimilarresultholdswhenthevarianceisestimatedbypluggingintheker-neldensityestimateoff.However,applicationofthesmoothedbootstraptotheStudentizedsamplequantileinthelattercase,whichweshalltermthesmoothedbootstrap-tmethod,succeedsinimprovingtheerrorordertoo(n−2/3),ifsecond-orderkernelsareusedinconjunctionwithappropriatelySMOOTHEDBO