Local asymptotic inference for nonparametric regression with censored survival data

We consider a penalised nonparametric estimation of the relative risk function in the Cox proportional hazards model for survival data with right censoring. We derive the convergence rate, functional Bahadur representation (FBR) and local asymptotic normality of the nonparametric estimator by using reproducing kernel Hilbert space, counting process and empirical process theory. The new theoretical results fill the gap in the smoothing splines literature for nonparametric estimation in survival models. Furthermore, we construct the corresponding local confidence intervals by the bootstrap method. Extensive simulation studies are conducted to validate the proposed method and compare with the Bayesian confidence intervals, and a data example from the Stanford heart transplant study is provided for illustration.