Nonparametric Estimation in the Cox Model

Abstract

Nonparametric estimation of the relative risk in a generalized Cox model with multivariate time dependent covariates is considered. Estimation is based on a penalized partial likelihood. Using techniques from Andersen and Gill, and Cox and O'Sullivan, upper bounds on rate of convergence in a variety of norms are obtained. These upper bounds match the optimal rates available for linear nonparametric regression and density estimation. The results are uniform in the smoothing parameter, which is an important step for the analysis of data dependent rules for the selection of the smoothing parameter.