Stone's theorem for distributional regression in Wasserstein distance

Clément Dombry (1, 2) , Thibault Modeste (3, 4, 5) , Romain Pic (1, 2)

1 UFC - Université de Franche-Comté
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
3 UCBL - Université Claude Bernard Lyon 1
4 ICJ - Institut Camille Jordan
5 PSPM - Probabilités, statistique, physique mathématique

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.

Mots clés: distributional regression; Wasserstein distance; non parametric regression; minimax rate of convergence

Domaines: Statistiques [math.ST]

Clément Dombry, Thibault Modeste, Romain Pic. Stone's theorem for distributional regression in Wasserstein distance. 2023. ⟨hal-03966752⟩