随机截断模型下的中心极限定理  

THE CENTRAL LIMIT THEOREM UNDER RANDOM TRUNCATION

作　　者：何书元[1] 

机构地区：[1]北京大学数学科学学院,北京100871

出　　处：《数学年刊（A辑）》2002年第3期345-354,共10页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.19971006)资助的项目.

摘　　要：在流行病学,生物统计学和天文学中常遇到随机截断数据.在随机截断下,人们关心的随机变量X被另一个随机变量Y干扰.只有当X≥Y时,才能观测到X和Y.在这个模型下,人们需要用截断数据估计X的分布函数F.本文证明,F的非参数最大似然估计Fn在下述意义下服从中心极限定理.对任何可测函数g（x）,n^（1/2）∫g（x）[dFn（x）-dF（x）]依分布收敛到均值为零方差为σ2的正态分布.从这个结果可以得出F的各种矩,特征函数等估计的渐近正态性.作为推论,还可以得到Fn在整个直线上的依分布收敛.我们的结果不要求X和Y的分布函数连续,得到的方差公式是简明的.Truncated data occur in many fields, for instance, in epidemiology, biometry and astronomy. In random truncation, the random variable X is interferred by another random variable Y. Both X and Y are observable only if X≥Y. A problem of interest is the estimation of the distribution function F of X with the truncated data. We show that the nonparametric MLE Fn of F obeys the central limit theorem in the sense that for measurable g(x), the integrals n^1/2 ∫ g(x)[dFn(x)- dF(x)] converges to a normal distributon with mean zero and variance σ2 as n tends to infinity. The results are useful in establishing the central limit theorems of various moment estimates, the empirical charactristic function and Fn on the whole line. In this paper, the theorems are obtained under very weak conditions in which the distributions of X and Y are left arbitrary and the derived asymptotic variance has a very simple from.

关 键 词：随机截断 乘积限估计 渐近正态性 

分 类 号：O211.6[理学—概率论与数理统计] O211.4[理学—数学]