Censored Partial Linear Models and Empirical Likelihood

Consider the partial linear model Yi=Xτiβ+g(Ti)+εi, i=1, …, n, where β is a p×1 unknown parameter vector, g is an unknown function, Xiʹs are p×1 observable covariates, Tiʹs are other observable covariates in [0, 1], and Yiʹs are the response variables. In this paper, we shall consider the problem of estimating β and g and study their properties when the response variables Yi are subject to random censoring. First, the least square estimators for β and kernel regression estimator for g are proposed and their asymptotic properties are investigated. Second, we shall apply the empirical likelihood method to the censored partial linear model. In particular, an empirical log-likelihood ratio for β is proposed and shown to have a limiting distribution of a weighted sum of independent chi-square distributions, which can be used to construct an approximate confidence region for β. Some simulation studies are conducted to compare the empirical likelihood and normal approximation-based method.