Estimating high‐dimensional additive Cox model with time‐dependent covariate processes

This paper is concerned with the estimation in the additive Cox model with time‐dependent covariates when the number of additive components p is greater than the sample size n. By combining spline representation and the group lasso penalty, a penalized partial likelihood approach to estimating the unknown component functions is proposed. Given the non‐iid nature of the log partial likelihood function and the nonparametric complexities of the component function estimation, it is challenging to analyze the theoretical properties of the proposed estimation. Through concentration inequities developed for martingale differences in the context of the additive Cox model, we establish nonasymptotic oracle inequalities for the group lasso in the additive Cox model with p=eo(n) under the compatibility and cone invertibility factors conditions on the Hessian matrix. An interesting and surprising aspect of our result is that the complexity of the component functions affects not only the approximation error but also the stochastic error. This is quite different from the additive mean models and suggests that the additive Cox model is more difficult to estimate than the additive mean models in high‐dimensional settings.