Asymptotic confidence bands for centered purely random forests

TL;DR

This paper develops explicit asymptotic confidence bands for centered purely random forests in multivariate nonparametric regression, introducing a new Ehrenfest variant that achieves optimal convergence rates.

Contribution

It introduces Ehrenfest centered purely random forests that attain minimax optimal rates, and provides a theoretical framework for confidence bands for both existing and new random forest types.

Findings

Ehrenfest random forests achieve minimax optimal rates.

Confidence bands are valid for both classical and Ehrenfest random forests.

Simulation examples illustrate the theoretical results.

Abstract

In a multivariate nonparametric regression setting we construct explicit asymptotic uniform confidence bands for centered purely random forests. Since the most popular example in this class of random forests, namely the uniformly centered purely random forests, is well known to suffer from suboptimal rates, we propose a new type of purely random forests, called the Ehrenfest centered purely random forests, which achieve minimax optimal rates. Our main confidence band theorem applies to both random forests. The proof is based on an interpretation of random forests as generalized U-Statistics together with a Gaussian approximation of the supremum of empirical processes. Our theoretical findings are illustrated in simulation examples.