大分位数与上端点的估计.pdf

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1、CHINESEJOURNALJune2014Vol.31No.3OFENGINEERINGMATHEMATICSdoi:10.3969/j.issn.1005-3085.2014.03.012ArticleID:1005-3085(2014)03-0424-11EstimatorsforaLargeQuantileandtheUpperEndpoint∗HELa-mei(CollegeofMathematics,SichuanUniversity,Chengdu610064)Abstract:Theestimationofextremev

2、alueindexisaprimaryprobleminextremevaluetheory.Inthispaper,basedonaPickands-typeestimatorfortheextremevalueindex,estimatorsforalargequantileandtheupperendpointofaprobabilitydistributionareestablished.Furthermore,theasymptoticpropertiesoftheseestimatorsarediscussed.Keyword

3、s:extremevalueindex;Pickands'estimator;largequantile;upperendpointClassication:AMS(2000)62G32;62G05;62G20CLCnumber:O211.4Documentcode:A1IntroductionLetX1;X2;···beindependentandidenticallydistributed(i.i.d.)randomvariables(r.v.)withacommondistributionfunction(d.f.)F(x),X1

4、;n≤X2;n≤···≤Xn;nstandfortheorderstatisticscorrespondingtor.v.sX1;X2;···;Xn.Supposethereexistsequencesan>0andbn∈RsuchthatlimP(Xn;n≤anx+bn)=G(x);x∈R;(1)n→∞where−1exp{−(1+ x)};1+x>0;̸=0;G(x)=−xexp{−e});x∈R;=0:Gisusuallycalledthegeneralizedextremevalue(GEV)distribution.I

5、f(1)holds,wesaythatthedistributionfunctionFisinthedomainofattractionofG(F∈D(G)).Therealparameteriscalledanextremevalueindex.Manyauthorshavediscussedtheestimationproblemontheextremevalueindex[1-15].Awell-knownestimatorof(∈R)isthePickandsestimator,whichwasproposedbyReceived

6、:21Nov2011.Biography:HeLamei(Bornin1971),Female,Ph.D.,AssociateProfessor.Accepted:24Apr2013.Researcheld:extremevaluetheoryandinformationfusion.Foundationitem:TheNationalNaturalScienceFoundationofChina(60874107;10771148);theOpeningFundofGeomathematicsKeyLaboratoryofSichu

7、anProvince(scsxdz2011006).NO.3HeLamei:EstimatorsforaLargeQuantileandtheUpperEndpoint425Pickands[13].Theestimatoris1Xn−m+1;n−Xn−2m+1;n^n=log(2)log2Xn−2m+1;n−Xn−4m+1;nm(n)foranysequenceofintegersm=m(n)satisfyingm(n)→∞;→0.Pickandsnprovedthatthisestimator^nisweaklyconsistent.

8、AlsothestrongconsistencyandasymptoticnormalitywereprovedbyDekkersandDeHaan[2].Oneproblemrelatedt