Oracle inequalities for the Lasso in the high-dimensional Aalen multiplicative intensity model

TL;DR

This paper develops a new high-dimensional Lasso-based estimator for the Cox model in counting processes, providing non-asymptotic oracle inequalities and leveraging advanced probabilistic tools.

Contribution

It introduces a data-driven weighted Lasso procedure for estimating the Cox model's intensity in high-dimensional settings, with theoretical guarantees.

Findings

Non-asymptotic oracle inequalities established

Utilizes empirical Bernstein's inequality for martingales

Provides theoretical bounds for high-dimensional Cox model estimation

Abstract

In a general counting process setting, we consider the problem of obtaining a prognostic on the survival time adjusted on covariates in high-dimension. Towards this end, we construct an estimator of the whole conditional intensity. We estimate it by the best Cox proportional hazards model given two dictionaries of functions. The first dictionary is used to construct an approximation of the logarithm of the baseline hazard function and the second to approximate the relative risk. We introduce a new data-driven weighted Lasso procedure to estimate the unknown parameters of the best Cox model approximating the intensity. We provide non-asymptotic oracle inequalities for our procedure in terms of an appropriate empirical Kullback divergence. Our results rely on an empirical Bernstein's inequality for martingales with jumps and properties of modified self-concordant functions.