Distribution-on-distribution regression considers the problem of formulating and estimating a regression relationship where both covariate and response are probability distributions. The optimal transport distributional regression model postulates that the conditional Fr\'echet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish the minimax rate of estimation of such a regression function, by deriving a lower-bound that matches the convergence rate attained by the Fr\'echet least squares estimator.

翻译：分布式分布式回归考虑的是制定和估计一种回归关系的问题,在这种关系中,共变和反应都是概率分布。最佳的运输分布式回归模型假设,响应分布的有条件的Fr\'echet值通过最佳运输图与共变分布挂钩。我们通过得出一个与Fr\'echet最小方形估计值所达到的趋同率相匹配的下限值来建立这种回归函数的最小估计速率。