Mathematical Sciences: Self-Consistent Estimators, Bootstrap and Censored Data

9510376 Ren摘要本研究涉及各种截尾数据的统计推断问题，包括右截尾数据、双截尾数据、区间截尾数据、左截断右截尾数据和双变量截尾数据。估计、检验和不完全日期的可信区间都在考虑之中。研究的目的之一是探索一种新的重抽样方法，称为杠杆Bootstrap，从而避免将通常的完整数据的估计和检验过程扩展到某些情况下的不完整数据或某些复杂模型的困难。到目前为止，为不完整数据制定的程序是在特别的基础上制定的。杠杆引导不依赖于这些即席扩展。本文主要研究杠杆Bootstrap在一些重要的不完全数据统计推断问题中的一致性问题，如经验似然推断、非参数似然置信带、线性回归模型、两个随机变量的独立性检验、一些半参数变点模型的检验、混合模型和有偏抽样问题。本研究的另一个目的是研究在使用各种删失数据的假设检验问题中，m of n Bootstrap的一致性，即重采样少于n个观测值。众所周知，当人们试图在半参数(受限非参数)假设下估计检验统计量的分布并忽略该假设施加的限制时，通常的非参数自举(n中的n自举)失败。由于截尾数据的检验统计量的分布通常是未知的，并且由于截尾变量的未知分布而非常复杂，因此研究通常自举的可能的补救措施以提供检验的临界值是非常有意义的。算例表明，在某些情况下，除了m of n Bootstrap外，没有明显的一般补救措施。这项研究的主要动机是使用不完全数据的统计推断问题，这些问题在医学和可靠性研究中经常遇到。特别是，在乳腺癌研究和艾滋病研究中分别出现了双重删失数据和区间删失数据。通常对完全数据的估计和检验方法不能直接适用于这种复杂的不完全数据，缺乏统计分析。这项研究的目的是探索一些通用的方法，这些方法将直接促进其他领域的研究，如医学。

9510376 Ren Abstract The proposed research is concerned with the statistical inference problems using various types of censored data, including right censored, doubly censored, interval censored, left truncated - right censored, and bivariate censored data. Estimation, tests and confidence intervals with incomplete date are all under consideration. One objective of the research is to investigate a new resampling method, called the Leveraged Bootstrap, so that the difficulty of extending the usual estimation and testing procedures for complete data to incomplete data in certain situations or certain complicated models can be avoided. So far, the procedures developed for incomplete data are constructed on an ad hoc basis. The leveraged bootstrap does not rely on these ad hoc extensions. The research will focus on the investigation of the consistency of the leveraged bootstrap in a number of important statistical inference problems with incomplete data, such as empirical likelihood inference, nonparametric likelihood confidence bands, linear regression models, tests of independence for two random variables, tests in some semiparametric change point models, mixture models and biased sampling problems. Another objective of the research is to investigate the consistency of the m out of n bootstrap, i.e., resampling fewer than n observations, in the hypothesis testing problems using various censored data. It is known that the usual nonparametric bootstrap (the n out of n bootstrap) fails when one tries to estimate the distribution of test statistics under a semiparametric (restricted nonparametric) hypothesis and ignores the restrictions imposed by the hypothesis. Since the distributions of test statistics for censored data are often unknown and very complicated due to the unknown distributions of the censoring variables, it is of great interest to investigate possible remedies of the usual bootstrap in order to provide the critical values for the tests. Examples show that in certain situations, there are no obvious general remedies except the m out of n bootstrap. This research is primarily motivated by statistical inference problems using incomplete data which are frequently encountered in medical and reliability studies. In particular, doubly censored data and interval censored data occur in breast cancer research and AIDS research, respectively. The usual estimation and testing procedures for complete data are not directly applicable to such complicated incomplete data, and statistical analysis is lacking. The objective of this research is to investigate some general methods which will directly facilitate research in other fields, such as medicine.