Open Access
2024 Uniform confidence band for optimal transport map on one-dimensional data
Donlapark Ponnoprat, Ryo Okano, Masaaki Imaizumi
Author Affiliations +
Electron. J. Statist. 18(1): 515-552 (2024). DOI: 10.1214/23-EJS2211


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.

Funding Statement

DP is financially supported by the Faculty of Science, Chiang Mai University, Thailand. RO was supported by Grant-in-Aid for JSPS Fellows (22J21512). MI was supported by JSPS KAKENHI (21K11780), JST CREST (JPMJCR21D2), and JST FOREST (JPMJFR216I).


We would like to show our gratitude to the associate editor and reviewers for their fruitful comments and suggestions.


Download Citation

Donlapark Ponnoprat. Ryo Okano. Masaaki Imaizumi. "Uniform confidence band for optimal transport map on one-dimensional data." Electron. J. Statist. 18 (1) 515 - 552, 2024.


Received: 1 July 2023; Published: 2024
First available in Project Euclid: 13 February 2024

Digital Object Identifier: 10.1214/23-EJS2211

Primary: 49Q22 , 62G15
Secondary: 62F40

Keywords: bootstrap , Confidence band , Optimal transport

Vol.18 • No. 1 • 2024
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