[Submitted] Stone’s theorem for distributional regression in Wasserstein distance
Published in , 2023
Recommended citation: Dombry, Clément, Modeste, Thibault, & Pic, Romain (2023). "Stone’s theorem for distributional regression in Wasserstein distance". arXiv. https://arxiv.org/pdf/2302.00975.pdf
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.
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