Details

Autor(en) / Beteiligte

• Ghosal, Promit
• Sen, Bodhisattva

Titel

Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing

Ist Teil von

• The Annals of statistics, 2022-04, Vol.50 (2), p.1012

Ort / Verlag

Hayward: Institute of Mathematical Statistics

Erscheinungsjahr

2022

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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.