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April 2022 Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing
Promit Ghosal, Bodhisattva Sen
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Ann. Statist. 50(2): 1012-1037 (April 2022). DOI: 10.1214/21-AOS2136


In this paper, we study multivariate ranks and quantiles, defined using the theory of optimal transport, and build on the work of Chernozhukov et al. (Ann. Statist. 45 (2017) 223–256) and Hallin et al. (Ann. Statist. 49 (2021) 1139–1165). We study the characterization, computation and properties of the multivariate rank and quantile functions and their empirical counterparts. We derive the uniform consistency of these empirical estimates to their population versions, under certain assumptions. In fact, we prove a Glivenko–Cantelli type theorem that shows the asymptotic stability of the empirical rank map in any direction. Under mild structural assumptions, we provide global and local rates of convergence of the empirical quantile and rank maps. We also provide a sub-Gaussian tail bound for the global L2-loss of the empirical quantile function. Further, we propose tuning parameter-free multivariate nonparametric tests—a two-sample test and a test for mutual independence—based on our notion of multivariate quantiles/ranks. Asymptotic consistency of these tests are shown and the rates of convergence of the associated test statistics are derived, both under the null and alternative hypotheses.

Funding Statement

The second author was supported by NSF Grant DMS-2015376.


The authors are extremely grateful to Peng Xu for creating the R-package (see [85]) for the computation of all the estimators studied in this paper. In particular, all of the plots in the paper are obtained from his R-package. The authors would like to thank Nabarun Deb, Adityanand Guntuboyina, Marc Hallin and Johan Segers for helpful discussions. The authors also acknowledge the many insightful comments by the two anonymous referees that helped improve the paper.


Download Citation

Promit Ghosal. Bodhisattva Sen. "Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing." Ann. Statist. 50 (2) 1012 - 1037, April 2022.


Received: 1 September 2020; Revised: 1 September 2021; Published: April 2022
First available in Project Euclid: 7 April 2022

Digital Object Identifier: 10.1214/21-AOS2136

Primary: 62G20 , 62G30
Secondary: 35J96 , 60F15

Keywords: Brenier–McCann’s theorem , convergence of subdifferentials of convex functions , Glivenko–Cantelli type theorem , Legendre–Fenchel dual , local uniform rate of convergence , semidiscrete optimal transport , testing mutual independence , two-sample goodness-of-fit testing

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 2 • April 2022
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