Multivariate weighted empirical likelihood MLE for the Cox model with various types of censored data

In analysis of survival data, the Cox model is one of the most widely used semi-parametric models, and we often encounter various types of censored survival data, such as right censored data, doubly censored data, interval censored data, partly interval-censored data, bivariate survival data subject to right censoring, etc. For complicated types of censored data, the studies of statistical inferences on the Cox model are very technical and challenging mathematically, thus up to now little work has been done. Among existing works, they either are not likelihood based, or are case-by-case based methods, which are not directly applicable to different types of censored data. In this article, we extend the concept of weighted empirical likelihood (WEL) from univariate case to multivariate case, and we apply it to the Cox model which leads to an estimation approach, called weighted maximum likelihood estimator (WMLE). Our proposed estimation procedure is applicable to various types of censored data under a unified framework, and some relevant asymptotic properties are established. Moreover, we obtain the WEL version of Cox’s partial likelihood estimator for various types of censored data, and provide stable computing algorithms for such estimator as well as for the WMLE. Some simulation results are presented.