na序列下经验分布函数的渐近正态性及其应用

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3Vol.26,No.320068JOURNALOFMATHEMATICALRESEARCHANDEXPOSITIONAug.,2006ArticleID:1000-341X(2006)03-0457-08Documentcode:AAsymptoticNormalityoftheEmpiricalDistributionunderNegativelyAssociatedSequencesanditsApplicationsLIYong-ming1,YANGShan-chao2(1.Dept.ofMath.,Shangrao

2、NormalCollege,Jiangxi334001,China;2.Dept.ofMath.,GuangxiNormalUniversity,Guilin541004,China)(E-mail:lym1019@163.com)Abstract:Bythewell-knownlargeandsmallblockspartingmethodfordependentsituations,weestablishtheasymptoticnormalityoftheEmpiricalDistributionFunctionunderNegatively

3、AssociatedSequences.Asitsapplicationinreliablityproblems,anaturalestimateFn(x)forthesurvivalfunctionF(x)=P(X>x)isproposed,andtheasymptoticnormalityof1n2[Fn(x)F(x)]isestablished.Keywords:NAsequences;empiricaldistribution;survivalfunction;asymptoticnormality.MSC(2000):62G05,62N

4、01CLCnumber:O212.71.IntroductionRandomvariablesfX1;


;Xn;n2garesaidtobenegativelyassociated(NA),ifforeverypairofdisjointsubsetsA1andA2off1;2;


;ng,Cov[g1(Xi;i2A1);g2(Xj;j2A2)]0;whereg1andg2areincreasingforeveryvariable(oraredecreasingforeveryvariable),suchthatcovarianceexi

5、sts.RandomvariablessequencefXj;j2Ngaresaidtobenegativelyassociatedifeverysubfamilyisnegativelyassociated.ThisdenitionwasintroducedbyJoag-DevandProschan[1].Becauseofitswideapplicationinreliabilitytheoryproblems,statisticalmechanics,probability/stochasticprocessesandstatistics,

6、thenotionofNArandomvariableshavereceivedmoreandmoreattentionrecently.LetX1;X2;


;Xnbeastrictlystationarynegativelyassociatedrandomvariablesequencewithdistribution(d.f.)F,andsetF(x)=P(X>x)forthesurvialfunction,whereXisdis-tributedaccordingtoF;Ofcourse,F(x)=1F(x).TheestimateFn

7、(x)istheempiricald.f.basedonX1;X2;


;Xn;andFn(x)=1Fn(x):TheestimationofthesurvivalfunctionF(x)andtheestablishmentofasymptoticallyopti-malpropertiesoftheproposedestimateFn(x)is,clearly,ofinterestonitsownright.Itisalsof(x)essentialinestimatingthehazardrateorfailurerateatx,r(x)

8、;wherer(x)=;x2R;withF(x)Receiveddate:2003-10-30Foundationitem