Stone's theorem for distributional regression in Wasserstein distance

2023/02/02 by Clément Dombry, Dombry, Clément, Thibault Modeste +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Risk and Portfolio Optimization #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2302.00975

openalex publication_date 2023/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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 ≥ 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.

Citations

• Consistent Nonparametric Regression
• One-dimensional empirical measures, order statistics, and Kantorovich transport distances
• Distributional Random Forests: Heterogeneity Adjustment and Multivariate\n Distributional Regression

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