Structured estimation for the nonparametric Cox model

Abstract

In this paper, we study theoretical properties of the non-parametric Cox proportional hazards model in a high dimensional non-asymptotic setting. We establish the finite sample oracle $l_{2}$ bounds for a general class of group penalties that allow possible hierarchical and overlapping structures. We approximate the log partial likelihood with a quadratic functional and use truncation arguments to reduce the error. Unlike the existing literature, we exemplify differences between bounded and possibly unbounded non-parametric covariate effects. In particular, we show that bounded effects can lead to prediction bounds similar to the simple linear models, whereas unbounded effects can lead to larger prediction bounds. In both situations we do not assume that the true parameter is necessarily sparse. Lastly, we present new theoretical results for hierarchical and smoothed estimation in the non-parametric Cox model. We provide two examples of the proposed general framework: a Cox model with interactions and an ANOVA type Cox model.