Learning and adaptive estimation for marker-dependent counting processes

TL;DR

This paper develops adaptive estimation methods for the intensity of marker-dependent counting processes, providing risk bounds and oracle inequalities that adapt to the smoothness and structure of the intensity function.

Contribution

It introduces a new empirical risk framework and aggregation algorithms with exponential weights for adaptive estimation in counting processes.

Findings

Risk bounds for empirical risk minimizers.

Oracle inequality for aggregation with exponential weights.

Adaptive estimators for anisotropic Besov spaces.

Abstract

We consider the problem of statistical learning for the intensity of a counting process with covariates. In this context, we introduce an empirical risk, and prove risk bounds for the corresponding empirical risk minimizers. Then, we give an oracle inequality for the popular algorithm of aggregation with exponential weights. This provides a way of constructing estimators that are adaptive to the smoothness and to the structure of the intensity. We prove that these estimators are adaptive over anisotropic Besov balls. The probabilistic tools are maximal inequalities using the generic chaining mechanism, which was introduced by Talagrand (2006), together with Bernstein's inequality for the underlying martingales.