# Boosted nonparametric hazards with time-dependent covariates

## TL;DR

This paper introduces a gradient boosting method for nonparametric hazard estimation with time-dependent covariates, providing theoretical guarantees and practical implementation insights.

## Contribution

It develops a convex representation of the nonparametric likelihood and a generic boosting algorithm, including an implementation with regression trees, with consistency and regularization analysis.

## Key findings

- The estimator is consistent under correct model specification.
- An oracle inequality is established for tree-based models.
- Step-size restriction prevents convergence issues due to risk curvature.

## Abstract

Given functional data from a survival process with time-dependent covariates, we derive a smooth convex representation for its nonparametric log-likelihood functional and obtain its functional gradient. From this, we devise a generic gradient boosting procedure for estimating the hazard function nonparametrically. An illustrative implementation of the procedure using regression trees is described to show how to recover the unknown hazard. The generic estimator is consistent if the model is correctly specified; alternatively, an oracle inequality can be demonstrated for tree-based models. To avoid overfitting, boosting employs several regularization devices. One of them is step-size restriction, but the rationale for this is somewhat mysterious from the viewpoint of consistency. Our work brings some clarity to this issue by revealing that step-size restriction is a mechanism for preventing the curvature of the risk from derailing convergence.