Weighted Empirical Likelihood

本研究的目的是研究一种新的似然函数，称为加权经验似然函数，在不同类型的不完全数据(包括右删失数据、双删失数据和区间删失数据)的生存分析中，构造各种重要的非参数或半参数统计推断问题的置信集和检验。对于不同类型的不完全数据，通过非参数极大似然估计的概率质量，建立了加权经验似然函数的统一形式。PI表明，对于上述各种类型的删失数据，一类相当一般的统计量的对数似然比的高阶展开式可以以统一的方式获得。因此，在调整后的n中n自举的帮助下，加权经验似然比可用于构造有效的可信区间或检验。该方法不需要对感兴趣统计量的收敛速度有精确的了解，这对区间截尾数据特别有吸引力。所有的初步研究表明，加权经验似然方法的理想性质包括精确度、效率、通用性和对感兴趣统计量的收敛速度的精确知识的独立性。在这项研究中，所考虑的问题包括：(A)加权经验似然在各种不完全数据的统计推断问题中的应用，包括与轮廓似然问题、加速寿命模型、比例风险模型和Logistic回归模型等相关的可信区间的构建和检验；(B)加权经验似然比可信区间的覆盖精度；(C)加权经验似然推断的效率；(D)与其他方法的比较(如果存在的话)。在医学随访和可靠性研究中，经常会遇到不完整的数据。近年来，随着在重要的临床试验和科学研究中出现的一些更复杂类型的不完全数据，如双删失数据、区间删失数据、截尾数据等，统计学家们越来越关注这些数据。例如，在最近的一项乳腺癌研究中遇到了双删失数据，在艾滋病研究中遇到了区间删失数据，在天文研究中遇到了双截断数据。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。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上的应用相对较少，而且大多是在正确删失的数据上。实例表明，对于复杂类型的不完全数据，如区间删失数据，对数似然比的极限分布的研究可能是相当困难的。在这种背景下，本项目旨在为医学和科学研究中出现的各种难题提供解决方案，本项目旨在开发基于一种新的似然函数的可靠统计方法，称为加权经验似然函数。

Proposal ID: DMS-0204182PI: Jian-Jian RenTitle: Weighted empirical likelihoodAbstract The objective of this research is to investigate the applications of a new likelihood function, called weighted empirical likelihood, in constructing confidence sets and tests for various important nonparametric or semiparametric statistical inference problems in survival analysis using different types of incomplete data, including right censored data, doubly censored data and interval censored data. Weighted empirical likelihood function is formulated in a unified form through the probability mass of the nonparametric maximum likelihood estimator for different types of incomplete data. The PI has shown that the high-order expansion of the log-likelihood ratio for a quite general class of statistics can be obtained in a unified way for various types of censored data aforementioned. Thus, with the help of the adjusted n out of n bootstrap, the weighted empirical likelihood ratio can be used to construct efficient confidence intervals or tests. The proposed procedure does not require the precise knowledge of the convergence rate of the statistic of interest, which is particularly appealing for interval censored data. All preliminary studies show that the desirable properties of weighted empirical likelihood method include accuracy, efficiency, generality, and independency of the precise knowledge of the convergence rate of the statistic of interest. In this research, the issues under consideration include: (a) Applications of weighted empirical likelihood in statistical inference problems with various types of incomplete data, including the construction of confidence intervals and tests associated with profile likelihood problems, accelerated life model, proportional hazards model, and logistic regression model, etc.; (b) Coverage accuracy of weighted empirical likelihood ratio confidence intervals; (c) Efficiency of weighted empirical likelihood inferences; (d) Comparison with alternative methods if they exist. 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 doubly censored data, 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, 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. 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 mostly on right censored data. Examples show that for complicated types of incomplete data, such as interval censored data, the investigation of the limiting distribution of log-likelihood ratio can be quite difficult. In this context, aiming to provide solutions for various difficult problems that arise in medical and scientific research, this project intends to develop reliable statistical methods based on a new likelihood function, called weighted empirical likelihood function.