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

9510376任摘要 本文研究了不同类型删失数据的统计推断问题，包括右删失、双删失、区间删失、左截断右删失和双变量删失数据。估计，测试和不完整的数据的置信区间都在考虑之中。研究的目的之一是研究一种新的估计方法，称为杠杆Bootstrap，从而避免了在某些情况下或某些复杂模型中将通常的完整数据估计和检验程序扩展到不完整数据的困难。到目前为止，为不完整数据制定的程序是在临时基础上构建的。利用的引导程序不依赖于这些临时扩展。本研究将集中在一些重要的不完全数据统计推断问题，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，半参数变点模型，混合模型和有偏抽样问题的检验中的杠杆自助法的一致性的调查。研究的另一个目的是研究m out of n bootstrap的一致性，即，在使用各种删失数据的假设检验问题中，众所周知，当人们试图在半参数（限制性非参数）假设下估计检验统计量的分布时，通常的非参数自举（n中取n的自举）会失败，并且忽略了该假设所施加的限制。由于截尾数据的检验统计量的分布往往是未知的，并且由于截尾变量的未知分布而非常复杂，因此研究通常的Bootstrap的可能补救措施以提供检验的临界值是非常有趣的。实例表明，在某些情况下，除了n中取m的引导外，没有明显的一般补救措施。 本研究的主要动机是统计推断问题，使用不完整的数据，这是经常遇到的医疗和可靠性研究。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程序不能直接适用于这种复杂的不完全数据，缺乏统计分析。本研究的目的是探讨一些通用的方法，这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.