We develop a statistical inference method for an optimal transport map between distributions on real numbers with uniform confidence bands. The concept of optimal transport (OT) is used to measure distances between distributions, and OT maps are used to construct the distance. OT has been applied in many fields in recent years, and its statistical properties have attracted much interest. In particular, since the OT map is a function, a uniform norm-based statistical inference is significant for visualization and interpretation. In this study, we derive a limit distribution of a uniform norm of an estimation error for the OT map, and then develop a uniform confidence band based on it. In addition to our limit theorem, we develop a bootstrap method with kernel smoothing, then also derive its validation and guarantee on an asymptotic coverage probability of the confidence band. Our proof is based on the functional delta method and the representation of OT maps on the reals.

翻译：我们针对实数域上分布间的最优传输映射，提出了一种具有一致置信带的统计推断方法。最优传输（OT）概念被用于衡量分布间的距离，而OT映射则用于构建该距离。近年来，OT已在多个领域得到应用，其统计特性也引起广泛关注。特别地，由于OT映射是一类函数，基于一致范数的统计推断对可视化和解释具有重要意义。在本研究中，我们推导了OT映射估计误差的一致范数的极限分布，并据此构建了一致置信带。除了极限定理外，我们还开发了基于核平滑的bootstrap方法，并进一步推导了其有效性证明以及置信带渐近覆盖概率的理论保证。我们的证明基于泛函delta方法以及实数域上OT映射的表示形式。