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 smoothed bootstrap method with 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：我们针对实数分布之间的最优传输映射，开发了一种基于一致置信带的统计推断方法。最优传输（OT）概念用于度量分布间的距离，而OT映射则用于构建该距离。近年来，OT已被广泛应用于多个领域，其统计特性引起了广泛关注。特别地，由于OT映射本身是一个函数，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，并基于此开发了一致置信带。除极限定理外，我们还提出了平滑自助法，并验证了该方法在置信带渐近覆盖概率上的有效性及保证。我们的证明基于泛函delta方法以及实数上OT映射的表示形式。