Nonparametric Maximum Likelihood Estimators for Multivariate Distributions and Related Inference Problems with Various Types of Censored Data

在多变量生存数据的分析中，我们经常会遇到一个向量的一个或多个分量由于截尾而不能完全观测的情况。这类数据的一种常见类型是，生存时间受到各种类型的审查，协变量变量，如治疗、性别等，完全可以观察到。这种类型的数据示例在关于骨髓移植、乳腺癌、艾滋病研究、心脏病等的重要医学研究中已经遇到。当随机向量的两个分量都是受单变量或双变量右审查的生存时间时，另一种常见类型的经审查的生存数据发生。在关于皮肤移植和肾脏疾病的医学研究中，已经遇到了这种类型的数据例子。本项目的目的是研究基于经验似然的多元分布函数的非参数极大似然估计(NPMLE)，为生存分析中几个重要的非参数和半参数推断问题提供解决方案和理论上的理解。该项目开发的统计方法将为多变量生存数据分析提供工具，这对医学研究、流行病学以及社会和行为科学都有直接影响。众所周知，基于多元生存数据的随机向量X的非参数分布估计是非常重要的，因为它为研究X的分量之间的关系提供了工具，在建模和检验等方面起着重要的作用。另外，在半参数模型假设下，如线性模型、COX模型、加速寿命模型等，为了研究协变量Z对生存时间T的影响，模型设置下的估计量往往可以表示为分布估计量的统计泛函，从而可以通过这些统计泛函的可微性和分布估计量的渐近性质来研究这些估计量的渐近性质。然而，大多数现有的带有上述生存数据的估计器都是临时的，并不是通常意义上的基于可能性的。而且，它们中的大多数要么包含负概率质量，要么依赖于核和带宽。自Owen(1988)以来，经验似然函数被普遍认为是非参数似然函数。基于经验似然的NPMLE的基本思想确保了它是一个合适的多元分布函数，这在实际中是可取的。但是，上述多元生存数据的基于经验似然的NPMLE在文献中一直没有得到认真的考虑，直到本项目的PI在最近的一篇论文中发现了带有删失生存数据的二元NPMLE的许多令人惊讶的良好性质，并研究了它对于离散协变量Z的渐近性质。在这个提出的经验似然中，加权经验似然、渐近方法和模拟将主要被使用，所考虑的问题包括：(A)对于各种类型的删失的多变量生存数据推导基于经验似然的NPMLE；(B)所得到的NPMLE的计算算法和渐近性质；(C)几种重要的半参数生存模型下统计泛函的推导和渐近性质。这个项目将提供一种通用的方法，用各种类型的删失多变量生存数据构造多变量分布估计量，这些数据通常具有理想的性质，并将为一些广泛使用的生存模型所涉及的几个重要和具有挑战性的统计推断问题提供解决方案。

In the analysis of multivariate survival data, we frequently encounter the situation where one or more components of a vector are not completely observable due to censoring. One common type of such data is when the survival time is subject to various types of censoring, and the covariate variables, such as treatments, gender, etc., are completely observable. Data examples of this type have been encountered in important medical research on bone marrow transplant, breast cancer, AIDS research, heart disease, etc. Another common type of censored survival data occurs when both components of the random vector are survival times that are subject to univariate or bivariate right censoring. Data examples of such type have been encountered in medical studies on skin grafts and kidney disease. The objective of this project is to study the empirical likelihood-based nonparametric maximum likelihood estimator (NPMLE) for multivariate distribution function with various types of censored multivariate survival data and to provide solutions and theoretical understanding of several important nonparametric and semi-parametric inference problems in survival analysis. The statistical methodology developed in this project will provide tools for multivariate survival data analysis, which has direct impact to medical research, epidemiology, and social and behavioral sciences. It is well-known that nonparametric distribution estimator of that of random vector X based on multivariate survival data is of great importance, because it provides tools to study the relation among the components of X and plays vital roles in modeling and testing, etc. It is also known that to study the effects of covariate Z on survival time T under semi-parametric model assumptions, such as linear models, the Cox model, accelerated life model, etc., the estimators under the model setting often can be expressed as statistical functional of the distribution estimator, thus the asymptotic properties of these estimators can be studied via the differentiability of these statistical functionals and the asymptotic properties of distribution estimators. However, most existing estimators with above mentioned survival data are ad hoc, and are not likelihood-based in the usual sense. Also, most of them either contain negative probability masses, or are kernel and bandwidth dependent. Since Owen (1988), the empirical likelihood function has been generally accepted as the nonparametric likelihood function. The essential idea of empirical likelihood-based NPMLE ensures that it is a proper multivariate distribution function, which is desirable in practice. But, the empirical likelihood-based NPMLE for aforementioned multivariate survival data had not been carefully considered in literature until a recent paper by the PI of this project, in which she discovered many surprisingly nice properties of the bivariate NPMLE with censored survival data, and studied its asymptotic properties for discrete covariate Z. In this proposed empirical likelihood, weighted empirical likelihood, asymptotic methods and simulations will be mainly used, and the issues under consideration include: (a) Derivation of empirical likelihood-based NPMLE for various types of censored multivariate survival data; (b) Computation algorithms and asymptotic properties of the resulting NPMLE; (c) Derivation and asymptotic properties of the statistical functionals under several important semi-parametric survival models. This project will provide a general methodology for constructing the multivariate distribution estimators with various types of censored multivariate survival data, which generally possesses desirable properties, and will provide solutions to several important and challenging statistical inference problems associated with some widely used survival models.