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We develop a statistical inference method for an optimal transport mapbetween distributions on real numbers with uniform confidence bands. Theconcept of optimal transport (OT) is used to measure distances betweendistributions, and OT maps are used to construct the distance. OT has beenapplied in many fields in recent years, and its statistical properties haveattracted much interest. In particular, since the OT map is a function, auniform norm-based statistical inference is significant for visualization andinterpretation. In this study, we derive a limit distribution of a uniform normof an estimation error for the OT map, and then develop a uniform confidenceband based on it. In addition to our limit theorem, we develop a smoothedbootstrap method with its validation and guarantee on an asymptotic coverageprobability of the confidence band. Our proof is based on the functional deltamethod and the representation of OT maps on the reals.

Uniform Confidence Band for Optimal Transport Map on One-Dimensional Data