Empirical likelihood and estimation in a partially linear varying coefficient model with right censored data

Abstract

In this paper, we study the empirical likelihood and estimation of parameters of interest in a partially linear varying coefficient model with right censored data. Two cases are considered: censoring is independent of the covariates, and censoring depends on the covariates. The bias-corrected empirical log-likelihood ratios for the regression parameters are presented, Wilks' theorem is proved, and thus the confidence regions of the regression parameters are constructed. The estimators of parametric and nonparametric components are constructed, their asymptotic distributions are obtained, and the consistent estimators of the asymptotic variances are also given. Furthermore, the partial profile empirical log-likelihood ratios for each component of the regression parameters and the coefficient functions are constructed, and show they are asymptotically chi-squared. The obtained results can be directly used to construct the confidence regions/intervals for the regression parameters and the pointwise confidence intervals for the coefficient functions. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared, undersmoothing of the coefficient functions is avoided, and existing data-driven methods can be effectively used to select the optimal bandwidth. Simulation studies compare the empirical likelihood method with one based on normal approximation and perform real data analysis.