Further Studies on Weighted Empirical Likelihood

摘要：自Owen(1988)以来，经验似然方法被用来构造基于非参数似然比的检验和置信集。研究表明，经验似然比推论具有与其他方法相当的准确性。然而，到目前为止，经验似然方法在删失数据中的应用相对较少，而且主要集中在正确删失的数据上。在这种背景下，PI在她2001年的论文中发现了一种新的似然函数，称为加权经验似然函数，它具有促进对各种类型的不完全数据的广泛类型的非参数和半参数统计量的研究的潜力，包括双删失数据、区间删失数据和部分区间删失数据。结果表明，加权经验似然函数可以看作是欧文经验似然函数在删失数据下的渐近形式。在过去的几年里，PI的研究表明，加权经验似然确实为处理复杂类型的删失数据的统计推断问题提供了一个有用的工具，否则这些数据很难处理。本课题的目的是进一步研究加权经验似然在各种截尾数据生存分析中几个重要的非参数或半参数统计推断问题中的应用。所考虑的问题包括：(1)将加权经验似然扩展到处理p维变量；(2)将加权经验似然进一步应用于与估计方程、轮廓似然和一些重要的生存模型相关的估计或模型评估问题；(3)加权经验似然比可信区间的覆盖精度；(4)与其他方法的比较。近年来，在重要的临床试验和科学研究中出现的一些更为复杂的不完全数据，如双删失数据、区间删失数据、部分区间删失数据、截断数据等，越来越受到统计学家的关注。例如，在最近的一项关于乳腺癌的研究中遇到了双重删失数据，在艾滋病研究中遇到了区间删失数据，在心脏病和糖尿病研究中遇到了部分区间删失数据，在天文学研究中遇到了双重截断数据。到目前为止，对这些更复杂类型的不完全数据的统计研究仍然普遍落后于对删失数据的统计研究，主要是因为它在技术上更具挑战性。该项目的预期结果是更好地理解了加权经验似然技术，并为在生物医学研究和流行病学研究中广泛使用的一些生存模型在观测数据被正确删失、双重删失、区间删失或部分区间删失时的几个重要统计推断问题提供了解决方案。

ABSTRACT:Since Owen (1988), the empirical likelihood method has been developed to construct tests and confidence sets based on nonparametric likelihood ratio. Studies have shown that the empirical likelihood ratio inferences are of comparable accuracy to alternative methods. However, so far the applications of empirical likelihood method to censored data are relatively few and are mainly focused on the right censored data. In this context, the PI discovered in her 2001-paper that a new likelihood function, called weighted empirical likelihoodfunction, has the potential to facilitate the research on a broad class of nonparametric and semiparametric statistics for various types of incomplete data, including doubly censored data, interval censored data, and partly interval-censored data. It is shown that weighted empirical likelihood may be viewed as the asymptotic version of Owen's empirical likelihood function for censored data. In the past few years, the PI's investigationshows that the weighted empirical likelihood indeed provides a useful tool to deal with statistical inference problems with complicated types of censored data which are otherwise quite difficult to handle. The objective of this current project is to further study the applications of weighted empirical likelihood in providing solutions for several important nonparametric or semiparametric statistical inference problems in survivalanalysis with various types of censored data. The issues under consideration include: (1) extension of the weighted empirical likelihood to deal with p-dimensional variables; (2) further applications of the weighted empirical likelihood to estimation or model assessment problems associated with estimating equations, profile likelihood and some important survival models; (3) coverage accuracy of weighted empirical likelihood ratio confidence intervals; (4) comparison with alternative methods.Incomplete data are frequently encountered in medical follow-up and reliability studies. Recently, statisticians are paying more attention to some more complicated types of incomplete data, such as doublycensored data, interval censored data, partly interval-censored data, truncated data, etc., as these data occur in important clinical trials and scientific research. For instance, doubly censored data were encountered in a recent study of primary breast cancer, interval censored data were encountered in AIDS research, partly interval-censored data were encountered in heart disease and diabetes studies,and doubly truncated data were encountered in astronomical research. Up to now, the statistical research on these more complicated types of incomplete data still generally lags behind that on right censored data,mainly because it is technically much more challenging. The expected results of this project are better understandingof the technique of weighted empirical likelihood, and providing solutions for several important statistical inference problems associated with some widely used survival models in biomedical research and epidemiological studies when observed data are right censored, doubly censored, interval censored, or partlyinterval-censored.