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

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