Abstract

This article investigates the problem of testing independence of two random vectors of general dimensions. For this, we give for the first time a distribution-free consistent test. Our approach combines distance covariance with the center-outward ranks and signs developed by Marc Hallin and collaborators. In technical terms, the proposed test is consistent and distribution-free in the family of multivariate distributions with nonvanishing (Lebesgue) probability densities. Exploiting the (degenerate) U-statistic structure of the distance covariance and the combinatorial nature of Hallin’s center-outward ranks and signs, we are able to derive the limiting null distribution of our test statistic. The resulting asymptotic approximation is accurate already for moderate sample sizes and makes the test implementable without requiring permutation. The limiting distribution is derived via a more general result that gives a new type of combinatorial noncentral limit theorem for double- and multiple-indexed permutation statistics. Supplementary materials for this article are available online.

摘要

本文研究了测试两个具有一般维度的随机向量的独立性问题。为此，我们首次给出了无分布的一致性测试。我们的方法将距离协方差与 Marc Hallin 及其合作者开发的中心向外等级和符号相结合。用技术术语来说，所提出的检验在具有非消失 (Lebesgue) 概率密度的多元分布族中是一致且无分布的。利用（退化的）U-距离协方差的统计结构和哈林中心向外等级和符号的组合性质，我们能够推导出我们的检验统计量的极限零分布。对于中等大小的样本，所得到的渐近近似已经是准确的，并且无需置换即可实现测试。极限分布是通过一个更一般的结果推导出来的，该结果为双索引和多索引置换统计提供了一种新型的组合非中心极限定理。本文的补充材料可在线获取。