Distance Covariance, Independence, and Pairwise Differences

Distance covariance (Székely, Rizzo, and Bakirov) is a fascinating recent notion, which is popular as a test for dependence of any type between random variables <i>X</i> and <i>Y</i>. This approach deserves to be touched upon in modern courses on mathematical statistics. It makes use of distances of the type |X−X′| and |Y−Y′|, where (X′,Y′) is an independent copy of (<i>X</i>, <i>Y</i>). This raises natural questions about independence of variables like X−X′ and Y−Y′, about the connection between cov(|X−X′|,|Y−Y′|) and the covariance between doubly centered distances, and about necessary and sufficient conditions for independence. We show some basic results and present a new and nontechnical counterexample to a common fallacy, which provides more insight. We also show some motivating examples involving bivariate distributions and contingency tables, which can be used as didactic material for introducing distance correlation.

距离协方差（Distance Covariance，由Székely、Rizzo及Bakirov提出）是近年来广受关注的新颖概念，常被用于检验随机变量<X>与<Y>之间任意类型的相依性。该方法理应在现代数理统计课程中得到介绍。它利用形如|X−X′|与|Y−Y′|的距离量，其中(X′,Y′)是(X,Y)的独立同分布副本。这自然引出了一系列相关问题：诸如X−X′与Y−Y′等变量的独立性、cov(|X−X′|,|Y−Y′|)与双中心化距离间协方差的关联，以及独立性的充要条件。本文给出若干基础结论，并针对一种常见谬误提出了一个全新且无需专业背景的反例，以提供更深入的理解。此外，本文还展示了若干涉及二元分布与列联表的启发性示例，可作为引入距离相关的教学素材。