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

University Of Maryland, College Park, College Park MD

Investigators

Abstract

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.