Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation
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DOI：
10.1080/01621459.2021.1923508
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发表时间：
2021-06-16
影响因子：
3.7
通讯作者：
Sen，Bodhisattva
中科院分区：
文献类型：
--
作者：
Deb，Nabarun;Sen，Bodhisattva
在本文中，我们提出了一个多维无分布非参数检验的一般框架，该框架基于用测度传递理论定义的多元秩的概念。与文献中其他现有的建议不同，这些多元秩与通常的一维秩共享许多有用的性质；最重要的是，这些排名是不受发行限制的。这一重要观察结果使我们能够设计出在零假设下完全不受分布影响的非参数检验。我们通过构造两个经典非参数问题的精确无分布检验来证明该方法的适用性：(I)随机向量之间相互独立性的检验，以及（II）多元分布相等性的检验。特别地，我们分别为测试场景(I)和（II）提出了距离协方差和能量统计的（多元）秩版本。在这两个问题中，我们导出了所提出的检验统计量的渐近零分布。我们进一步表明，我们的测试对于所有固定的替代方案都是一致的。此外，拟议的测试在计算上是可行的，并且在对潜在分布的最小假设下定义良好（例如，它们不需要任何力矩假设）。我们还通过广泛的模拟证明了这些程序的有效性。在分析这些方法的理论性质的过程中，我们利用Stein的交换对方法证明了测量输运理论和置换统计极限理论中的一些新结果，这些结果可能具有独立的意义。
In this article, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure transportation. Unlike other existing proposals in the literature, these multivariate ranks share a number of useful properties with the usual one-dimensional ranks; most importantly, these ranks are distribution-free. This crucial observation allows us to design nonparametric tests that are exactly distribution-free under the null hypothesis. We demonstrate the applicability of this approach by constructing exact distribution-free tests for two classical nonparametric problems: (I) testing for mutual independence between random vectors, and (II) testing for the equality of multivariate distributions. In particular, we propose (multivariate) rank versions of distance covariance and energy statistic for testing scenarios (I) and (II), respectively. In both these problems, we derive the asymptotic null distribution of the proposed test statistics. We further show that our tests are consistent against all fixed alternatives. Moreover, the proposed tests are computationally feasible and are well-defined under minimal assumptions on the underlying distributions (e.g., they do not need any moment assumptions). We also demonstrate the efficacy of these procedures via extensive simulations. In the process of analyzing the theoretical properties of our procedures, we end up proving some new results in the theory of measure transportation and in the limit theory of permutation statistics using Stein’s method for exchangeable pairs, which may be of independent interest.