-On asmptotic behavior for weighted sample quantiles英文文献资料

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1、ONASYMPTOTICBEHAVIOROFWEIGHTEDSAMPLEQUANTILESK.Borovkov1,H.Dehling2andD.Pfeifer31Dept.ofStatistics,UniversityofMelbourneParkville3052,Australia2DepartmentofMathematics,UniversityofGroningenP.O.Box800,9700Groningen,TheNetherlands3Institutf•urMathematischeStochastik,

2、Universit•atHamburgBundesstr.55,20146Hamburg,GermanyWestudythelimitbehaviorofL-statisticswiththe`deltatype'weightsequences,ofwhichtheO-statisticsandusualkernelquantileestimatorsarespecialcases.Deviationsofthesestatisticsfromthesamplequantilesareshowntohave(afterpro

3、pernormaliza-tion)Gaussianlimitdistributions.Weevaluatethemaintermoftheasymptoticsofthecovarianceofthisdeviationwiththecorrespondingsamplequantile.Thisgivesalsotheasymptoticrelativedeciencyofsamplequantilesw.r.t.ourL-statistics.Keywords:samplequantile,quantileesti

4、mator,L-statistic,quantileprocess,rela-tiveeciency,secondorderproperties.1.Introduction.LetX1;:::;Xnbeasampleofrandomvariableswithahypotheticalcontin-uousdistributionfunctionF=F(n)(dependingingeneralcaseonn,sothatweareinthetriangulararrayschemesetup),X(1)X(2):::

5、X(n)betheorderstatisticsofthesample.LastdecadealotofpapersweredevotedtothestudyofvariousversionsofL-statisticsXnXn(n)(1:1)vjX(j);vj=vj;j=1;:::n;vj=1;j=1j=1havingthepropertythat,forany">0,Xjvjj!0asn!1jj=ntj>"forsomexedt2(0;1).Suchstatisticsareintendedforestimatin

6、gthevalueofthequantilefunction(1:2)Q(t)=Q(n)(t)=F1(t):=supfx:F(x)tg;0t1;1atthispointt,andinregularcasetheyturntobesuperiortothesamplequantiles,i.e.thevaluesoftheempiricalquantilefunction(e.q.f.)1j1jQn(t)=Fn(t)=X(j)fort<;j=1;:::;n;nnFnbeingtheempiricaldistrib

7、utionfunction(e.d.f.)ofthesampleX1;:::;Xn.Thisobservationwassupportedbybothexperimentaldata(seee.g.discussionsandsimulationdatainKaighandCheng[16],KaighandDriscoll[17],SheaterandMarron[24],SteinbergandDavis[27],Stewart[28],andYang[30])andtheoreticalconsiderationsfo

8、rthecaseofindependentidenticallydistributed(i.i.d.)X's(hereonecouldmentioninadditiontheworksbyAzzalini[1],DriscollandKaigh[6],Kaigh[15],Falk[7{10