On universally consistent and fully distribution-free rank tests of vector independence
On universally consistent and fully distribution-free rank tests of vector independence
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DOI：
10.1214/21-aos2151
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发表时间：
2020-07
期刊：
影响因子：
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通讯作者：
Hongjian Shi;M. Hallin;M. Drton;Fang Han
中科院分区：
文献类型：
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作者：
Hongjian Shi;M. Hallin;M. Drton;Fang Han
在过去的十年里，等级相关性已经发现了许多创新的应用。特别是，适当的等级相关性已被用于对随机变量对之间的独立性的一致性检验。随着测试变得无分布，使用RANK对连续数据特别有吸引力。然而，传统的排名概念依赖于排序数据，因此与单变量观测相联系。因此，长期以来，人们一直不清楚如何构建随机向量之间无分布但一致的独立性测试。这就是本文所要解决的问题，在这篇文章中，我们设计了一个通用的框架，用于设计相关性度量，这些度量给出的多元独立性检验不仅是一致的和无分布的，而且我们还证明了它在统计上是有效的。我们的框架利用了最近引入的中心-外向排名和符号的概念，这是对传统排名的多变量概括，并采用了一个通用的标准形式来衡量依赖性，其中包含了许多流行的例子。在统一的研究中，我们得到了独立条件下中心向外秩基检验统计量的一般渐近表示，并将经典的Hájek渐近表示结果推广到多元情形。这种表示法允许直接计算极限零分布，并促进了局部功率分析，该局部功率分析通过首次建立基于中心向外排序的检验在二次均值可微选方案类内的根-n邻域上的非平凡功率，从而为中心向外方法提供了强有力的支持。
Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent tests of independence between pairs of random variables. Using ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result, it has long remained unclear how one may construct distribution-free yet consistent tests of independence between random vectors. This is the problem addressed in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, and adopts a common standard form for dependence measures that encompasses many popular examples. In a unified study, we derive a general asymptotic representation of centeroutward rank-based test statistics under independence, extending to the multivariate setting the classical Hájek asymptotic representation results. This representation permits direct calculation of limiting null distributions and facilitates a local power analysis that provides strong support for the centeroutward approach by establishing, for the first time, the nontrivial power of center-outward rank-based tests over root-n neighborhoods within the class of quadratic mean differentiable alternatives.