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.(2017) and Hallin et al.(2021). 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 L_2-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.

翻译：在本文中,我们研究了使用最佳运输理论界定的多变等级和四分位数,并以Chernozhukov等人(2017年)和Hallin等人(2021年)的工作为基础,我们研究了多变等级和四分位数功能及其经验性对应功能的定性、计算和属性,根据某些假设,我们将这些经验性估计与其人口版本统一一致。事实上,我们证明是一种格利文科-Cantelli型的理论,显示经验性排名图在任何方向上的无变化性稳定性。在温和结构假设下,我们提供了经验性定量和级图的全球和地方趋同率。我们还为经验性四分位数函数的全球L_2损失提供了一种亚加西尾。此外,我们提议根据我们关于多变孔数/等级的观念,调整无参数性多变数非参数非参数非参数性非参数性非参数性非参数性测试 -- -- 一种两次抽样测试和相互独立的测试。这些测试显示这些测试的一致性。这些测试的典型一致性和等级图的趋同率都是在替代试验中得出的。