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.

翻译：我们将著名的Stone定理推广至分布回归框架。具体而言，我们证明满足Stone定理条件的局部概率权重加权经验分布能够提供条件分布的一致相合估计，该估计的误差由p阶Wasserstein距离（p≥1）度量。进一步地，对于p=1的情形，我们确定了特定分布类上的极小极大收敛速率。最后，我们给出了这些结果在条件尾部期望与概率加权矩估计中的若干应用。