The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the $L^2$ norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.

最优交通理论在许多经济研究领域的兴趣急剧增加，例如最优匹配理论和计量经济学识别。一个特别有价值的工具，由于它方便地表示为凸函数的梯度，是 Brenier 映射：作为 Monge-Kantorovich 最优运输问题的优化器获得的匹配，欧几里得距离作为成本函数。尽管 Brenier 地图很受欢迎，但其统计特性尚未完全确定，这阻碍了其在估计和推理中的实际应用。本文朝着这个方向迈出了第一步，通过半对偶 Monge-Kantorovich 问题推导了 Brenier 映射潜力的简单插件估计器的收敛速度。依赖于平滑经验过程收敛的经典结果，表明如果其中一个概率度量满足 Poincaré 不等式，则该插件估计器在核密度估计器的最小最大收敛率下收敛到其人口对应物的标准差。在电位的归一化下，结果延伸到收敛 $L^2$ 范数，而自动满足 Poincaré 不等式。本文的主要数学贡献是对半对偶 Monge-Kantorovich 问题的第二个变体的分析，该问题具有独立的兴趣。