Distance Covariance, Independence, and Pairwise Differences

TL;DR

This paper explores the properties and implications of distance covariance for testing independence between variables, providing new insights, counterexamples, and educational examples to deepen understanding of the method.

Contribution

It offers new theoretical results, clarifies misconceptions, and presents illustrative examples related to distance covariance and independence testing.

Findings

Basic properties of distance covariance analyzed

Counterexample to common fallacy provided

Examples with bivariate distributions and tables included

Abstract

(To appear in The American Statistician.) Distance covariance (Sz\'ekely, Rizzo, and Bakirov, 2007) is a fascinating recent notion, which is popular as a test for dependence of any type between random variables $X$ and $Y$. 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 $(X,Y)$. 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.