RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS.

IF 3.2 1区 数学 Q1 STATISTICS & PROBABILITY
Annals of Statistics Pub Date : 2024-02-01 Epub Date: 2024-03-07 DOI:10.1214/23-aos2339
Yeqing Zhou, Kai Xu, Liping Zhu, Runze Li
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引用次数: 0

Abstract

To test independence between two high-dimensional random vectors, we propose three tests based on the rank-based indices derived from Hoeffding's D, Blum-Kiefer-Rosenblatt's R and Bergsma-Dassios-Yanagimoto's τ*. Under the null hypothesis of independence, we show that the distributions of the proposed test statistics converge to normal ones if the dimensions diverge arbitrarily with the sample size. We further derive an explicit rate of convergence. Thanks to the monotone transformation-invariant property, these distribution-free tests can be readily used to generally distributed random vectors including heavily tailed ones. We further study the local power of the proposed tests and compare their relative efficiencies with two classic distance covariance/correlation based tests in high dimensional settings. We establish explicit relationships between D,R,τ* and Pearson's correlation for bivariate normal random variables. The relationships serve as a basis for power comparison. Our theoretical results show that under a Gaussian equicorrelation alternative, (i) the proposed tests are superior to the two classic distance covariance/correlation based tests if the components of random vectors have very different scales; (ii) the asymptotic efficiency of the proposed tests based on D,τ* and R are sorted in a descending order.

基于秩的指数,用于测试两个高维向量之间的独立性。
为了检验两个高维随机向量之间的独立性,我们提出了三种检验方法,分别基于从霍夫丁的 D、布卢姆-基弗-罗森布拉特的 R 和贝格斯马-达西奥斯-扬纳基莫托的τ* 得出的基于秩的指数。在独立性的零假设下,我们证明了如果维数随样本量任意发散,所提出的检验统计量的分布会收敛到正态分布。我们进一步推导出了明确的收敛率。得益于单调变换不变的特性,这些无分布检验可以很容易地用于一般分布的随机向量,包括重尾向量。我们进一步研究了所提出检验的局部功率,并比较了它们与两种基于距离协方差/相关性的经典检验在高维环境下的相对效率。我们在双变量正态随机变量的 D、R、τ* 和皮尔逊相关性之间建立了明确的关系。这些关系可作为功率比较的基础。我们的理论结果表明,在高斯等相关性替代条件下,(i) 如果随机向量的分量具有非常不同的尺度,所提出的检验优于基于距离协方差/相关性的两种经典检验;(ii) 基于 D、τ* 和 R 所提出的检验的渐进效率按降序排列。
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来源期刊
Annals of Statistics
Annals of Statistics 数学-统计学与概率论
CiteScore
9.30
自引率
8.90%
发文量
119
审稿时长
6-12 weeks
期刊介绍: The Annals of Statistics aim to publish research papers of highest quality reflecting the many facets of contemporary statistics. Primary emphasis is placed on importance and originality, not on formalism. The journal aims to cover all areas of statistics, especially mathematical statistics and applied & interdisciplinary statistics. Of course many of the best papers will touch on more than one of these general areas, because the discipline of statistics has deep roots in mathematics, and in substantive scientific fields.
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