Empirical likelihood for partially linear proportional hazards models with growing dimensions

Empirical-likelihood-based inferences for the linear part in a partially linear Cox’s proportional hazards model are investigated. It was shown in some previous studies, for some related but different semiparametric models, that if there is no bias correction, the limit distribution of the empirical likelihood ratio statistic is not a standard chi-square distribution. In some previous studies, the bias correction is achieved by subtracting a conditional expectation of a predictor from itself. In proportional hazards models, the situation is different and it is not clear how to do so. Motivated from the form of the asymptotic variance of the parameters, the bias-corrected empirical likelihood ratio is proposed, with a standard χ2χ2 limit. The demonstrated asymptotics even apply to models with growing dimensions. For computational simplicity, we use polynomial splines to approximate the nonparametric component so that the computations involved are similar to those for the parametric model. Some simulations are carried out to study the performance of bias-corrected empirical likelihood ratio.