LOCAL ASYMPTOTIC PROPERTIES OF HAZARD RATE ESTIMATORS FOR TRUNCATED AND CENSORED DATA

1 IntroductionLet (X, T, Y) denote random variables where X is the variab1e of interest, called the lifetimevariab1e, with distribution function (d.f.) F; T is the random 1eft truncation time with arbitraryd.f G and Y is the random right censoring time with arbitrary d.f. H. It is assumed that Xis independent Of (T, Y), bnt T and Y may be dePendent and, Without 1oss of generality thatthey are nonnegative. In the random 1eft truncation and right censoring (LTRC) model oneobserves (Z, T…     

LAWS OF THE ITERAFED LOGARITHM FORTRIMMED SUM

1.IntroductionLetX,X1,X2,''beasequenceofn0n-degenerateindependelltidenticallydistributedra-n-domVarableswithcommondistributionfunctionF,andletXl,'',X2beanarrangementofXl,'Xnindecreasingorderofmagnitude,i.e.IX:l2''2IX2I.SetS;=Z:. ,X;.Forx>0,defineandforx=0,setG(O)=Q(O)=P(IXI>O),K(0)=0-DeAneLetrnbeanynon-decreasingsequenceofilltegerssuchthatwheref>0andwewritel2nforloglogn.By(1.1)and(l.2),forsufficientlylargentwecandeflneasequenceb.byP.S.Griffinl1Jobtainedthefollowingtheorem:R…     

ASYMPTOTIC PROPERTIES OF L_1-NORM KERNEL ESTIMATOR OF THE CONDITIONAL MEDIAN

Let(X,Y) be a pair of R~d×R~1-valued random variables.In thispaper we investigate the asymptotic properties of the L_1-norm kernel estimator ofthe conditional median function of Y on X.Under appropriate regularity condi-tions,asymptotic normality and the optimal rates of convergence n~((-1)/(2+d))and(n~(-1)log n)~(1/(2+d)) in the L~q(1(?)q<∞)-and L~∞-norms restricted to a compactset,respectively,are obtained.Our study shows that this estimator and the well-known Nadaraya-Watson's kernel estimator of the conditional mean function of Yon X have the same asymptotic properties.     

THE RATE OF UNIFORM CONVERGENCE OF THE SURVIVAL FUNCTION ESTIMATOR FOR TRUNCATED AND CENSORED DATA

1 Illtroduction and Main ResultsIn medical follow-up or engineering life testing studies one may not be able to observethe variable of inferest, referred to hereafter as the lifetime. Among the differellt forms inwhich incomplete data appears right censoring and left truncation are two common ones. Lefttruncation mad occur if the time origin of the lifetime precedes the time origin of the study.Only subjects that fail after the start of the study are being followed, otherwise they are lefttru…     

BAHADUR'S REPRESENTATION FOR NEAREST NEIGHBOR MEDIAN ESTIMATES: THE FIXED DESIGN CASE

1.IntroductionConsiderthefixeddesignnonparametricregressionmodelK=g(xi) ei,i21,(1.l)whereg(x)isanunknownregressionfunctiondefinedontheinterval[0,if,{xi,iZ1}arefixeddesignpoilltsdrawnfromtheinterval[0,1],{ei,i21}arei.i.d.randomvariablesfromapopulationhavingunknownsymmetricdistributionF(x)withsymmetricpoint0andK(i21)istheobservationatxi(i21).Forestimatingtheunknownregressionfunctiong(x),takethefirstnobservationsYI,'',Ye.WithoutlossofgeneralityjletalSxZS''5xu.LetD.i(x)denotethei-thnearestn…     

ON THE ONE-SIDED LOGARITHMIC LAW FOR ARRAYS

1.IntroductionLet{X.,n2l}beasequenceofindependentandidenticallydistributed(i.i.d.)randomvariables.TheHartmanandWintner,sllllawoftheiteratedlogarithmstatesthatifEXI~0andEX7=1Moreover,itisprovedin[2]and[3]that(1)holdsifandonlyifEXI~0andEX7=1.Nowlet{Xnk,k=1,2,''tntn~1,2,'.}beanarrayofi.i.d.randomvariables.FOreveryn21setS.~Z:--,X.k'ThealmostsureconvergenceofSahasbeenstlldiedbyHu,MoriczandTaylor[41,HuandWeb.rls]andQi[6].HuandWeber[5]showedundertheconditionsEXll=0,EX7,=1andEXt,相似文献