Abstract
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 -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.
Acknowledgments
The authors are extremely grateful to Peng Xu for creating the R-package https://github.com/Francis-Hsu/testOTM (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.
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. https://doi.org/10.1214/21-AOS2136
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