Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data and continuous covariate

摘要

Recently, Ren and Riddlesworth (2014) derived the empirical likelihood-based bivariate nonparametric maximum likelihood estimator (BNPMLE) $hat{F}_n (t, z)$ for the bivariate distribution function $F_0 (t, z)$ of survival time $T$ and covariate variable $Z$ based on bivariate data where $T$ is subject to right censoring. They showed that such BNPMLE $hat{F}_n (t, z)$ is a consistent estimator of $F_0 (t, z)$ when variable $Z$ is discrete. Despite all nice properties of the BNPMLE $hat{F}_n (t, z)$ shown in Ren and Riddlesworth (2014), in this article we show that surprisingly such $hat{F}_n (t, z)$ is not a consistent estimator when the covariate variable $Z$ is continuous. On the other hand, interestingly our simulation studies suggest that some remedy adjustments on $hat{F}_n (t, z)$ based on the usual empirical likelihood treatments and the censoring mechanism may provide consistent estimators for $F_0 (t, z)$ with continuous covariate $Z$.