Stone's theorem for distributional regression in Wasserstein distance

TL;DR

This paper extends Stone's theorem to distributional regression using Wasserstein distance, establishing universal consistency and convergence rates, with applications in tail expectation and probability weighted moments.

Contribution

It introduces a framework for distributional regression with Wasserstein distance, proving consistency and minimax rates, which was not previously established.

Findings

Weighted empirical distributions are universally consistent for conditional distributions.

Minimax convergence rates are derived for p=1 Wasserstein distance.

Applications include estimation of tail expectations and probability weighted moments.

Abstract

We extend the celebrated Stone's theorem to the framework of distributional regression. More precisely, we prove that weighted empirical distribution with local probability weights satisfying the conditions of Stone's theorem provide universally consistent estimates of the conditional distributions, where the error is measured by the Wasserstein distance of order p $\ge$ 1. Furthermore, for p = 1, we determine the minimax rates of convergence on specific classes of distributions. We finally provide some applications of these results, including the estimation of conditional tail expectation or probability weighted moment.