Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing
Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing
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DOI：
10.1214/21-aos2136
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发表时间：
2019-05
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影响因子：
--
通讯作者：
Promit Ghosal;B. Sen
中科院分区：
文献类型：
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作者：
Promit Ghosal;B. Sen
在本文中，我们研究了用最优运输理论定义的多变量等级和分位数，并建立在Chernozhukov等人(2017)和Hallin等人(2021)工作的基础上。我们研究了多元秩函数和分位数函数及其经验对应函数的特征、计算和性质。在一定的假设条件下，我们得到了这些经验估计与它们的总体版本的一致一致性。事实上，我们证明了一个Glivenko-坎特利型定理，它表明了经验秩映射在任何方向上的渐近稳定性。在温和的结构假设下，我们给出了经验分位数和等级映射的全局和局部收敛速度。我们还给出了经验分位数函数的整体L_2损失的亚高斯尾界。此外，基于我们的多变量分位数/等级的概念，我们提出了调谐无参数的多变量非参数检验--两样本检验和相互独立性检验。在零假设和替代假设下，证明了这些检验的渐近相合性，并导出了相关检验统计量的收敛速度。
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.