随机删失模型中的渐近理论

授予学位单位：北京大学 授予时间：1996年7月
二级学科名称：概率论与数理统计
指导教师：郑忠国
博士论文题目：随机删失模型中的渐近理论

摘 要

本文第一章第1.1节使用Kaplan-Meier估计渐近方差的刀切估计,建立了学生化Kaplan-Meier估计的Edgeworth展开与被估计的Edgeworth展开.证明了两种Edgewor-th展开逼近其分布的渐近速度均为 . 第一章第1.2节建立了Tsai等人的乘积限估计的U-统计量表示， 利用该表示建立了该估计的Berry-Essen不等式，而且利用Gijbels等(1993)的定理1(c)得到了估计的r(≥2）阶绝对矩不等式与strassen型重对数律。 设S(s, t)是二元生存函数,是Campbell与Foldes(1982)所提出的二元乘积限估计,第一章第1.3节通过将 表成U-统计量加上具有所需性质的余项,利用U-统计量的Berry-Essen定理,建立了二元乘积限估计的Berry-Essen不等式。
第二章第2.1节给出了随机删失下概率密度估计的L1-矩不等式，P(≥2)阶绝对矩不等式,与一个概率不等式，建立了概率密度的光滑bootstrap逼近定理，并证明了概率密度bootstrap估计的方差几乎处处收敛到概率密度估计的渐近方差。 第二章第2.2节构造了失效率的一种核估计， 并研究了该估计的弱收敛速度，一致强相合性收敛速度，渐近表示与渐近正态性等问题。 第二章第2.3节定义了生存分布的一种平均型泛函的估计， 利用点过程理论与鞅方法证明了该估计的渐近正态性，此外，还给出了一个均方误差不等式和一个概率不等式。
第三章研究了随机删失非参数回归模型与半参数回归模型。 在第3.1节， 我们分别就删失分布已知与未知两种情况构造了非参数回归模型中回归函数的一种加权核估计，并研究了它的一些收敛性质。 对固定设计半参数回归模型 第3.2节也是分别就分布已知与未知两种情况，定义了参数 与回归函数的估计证明了它们均具有强相合性与 阶平均相合性。

Title of the Dr.Degree’s thesis: Asympototic Theory for Random Censorship
Model

Introduction of the Author: Wang Qi-Hua, was born on July 18, 1963. Under
the guidence of Prof. Zheng Zhong-Guo, he was awarded the Dr. degree of
science at Peking University on July 10,1996. Now, he continues his
research works in the field of Survival Analysis.

Kew Words: Product-Limit Estimator, Functional of Product-Limit, Censor-
ship Regression

Abstract

In Section 1.1 of Chapter 1, the jackknife estimte of the asymptotic
variance of Kaplan-Meier (KM) estimator is employed, and the Edgeworth
expansions for the studentized Kaplan-Meier estimator are established.It
is shown that the Edgeworth expansions are asympototic close to the exact
distribution of the KM estimator with remainder under some mild
conditions.

In Section 1.2 of Chapter 1, it is shown that the product-limit (PL)
estimator from left truncated and right censored data, which is defined
by Tsai et al(1987), may be approximated within a sufficient degree of
accuracy by a U-statistic. In this way, Berry-Essen theorem for U-statis-
tic does carry over to the PL estimator. Moreover, a direct application
of Theorem 1(c) of Gijbels et al(1993) yields the rth(r>1)
order absolute moment inequality and functional law of iterated logarithm
law for the error of the PL estimator.

Let S(s,t) be the bivariate survival function. Let be the
bivariate PL estimator proposed by Campbell and Foldes(1982). In Section
1.3 of Chapter 1, the Berry-Essen inequality for is established
by expressing as a U-statistic, which admids a term
Edgeworth expansion, plus some remainders with sufficient accuracy.

In Section 2.1 of Chapter 2, the kernel density estimate from
random censored data is investigated further. L1-moment, p( 2) order
absolute moment inequalities and a probability inequality for the devia-
tion of are obtained, respectively. The sufficient conditions
for the smoothed bootstrap approximation of to be valid are establi-
shed. Moreover, it is shown that the variance of the smoothed bootstrap
estimator converges to the asympototic variance of almost
surely.

In Section 2.2 of Chapter 2, a nonparametric hazard estimator is intro-
duced. Weak convergence,strong uniformly consistency of the proposed esti-
mator are investigated on a bounded intervals, respectively. An asym-
ptiotic representation is also given, and the asympototic representation
is used to prove asymptotic normality of the hazard estimator.

In Section 2.3 of Chapter 2,a class of mean type functional of KM esti-
mator are investigated. Counting process martingal methods are used to
show the asympototic normality, and establish a mean square error inequa-
lity and a probability inequality of them without the assumption that F,
G are continuous, where F, G are survival time distribution and censoring
time distribution respectively.

In Section 3.1 of Chapter 3, weighted kernel estimators of regression
function in the fixed design nonparametric model are constructed, respec-
tively, in the two cases that censoring distribution is known and unknown.
And some convergent properties of them are also investigated under
different conditions.

For the fixed design semiparametric regression model

Section 3.2 of Chapter 3 defines the estimators of unknown parametric
and regression function g( ), when the observations are randomly censored
on the right in the two cases:Censoring distribution is known and unknown,
respectively. The sufficient conditions under which these estimators are
strong consistent and pth(p 2) mean consistent are also established.

Research Achivements:

1. Wang Qi-Hua, Chinese Science, 39, 163(1986).
2. Wang Qi-Hua, Chinese Science Bulletin, 40, 632(1995).
3. Wang Qi-Hua, Acta Math. Scientia, 15, 123(1995).
4. Wang Qi-Hua, Chinese Journal of Probability and Statistics, 10,164(1994).
5. Wang Qi-Hua, Chinese Science Bulletin, 41, 3(1996).
6. Wang Qi-Hua, Acta Math. Scientia, 16, 376(19940).