Distance Covariance, Independence, and Pairwise Differences

Abstract

(To appear in The American Statistician.) Distance covariance (Székely, 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 $|X-X^{\prime }|$ and $|Y-Y^{\prime }|$, where $(X^{\prime },Y^{\prime })$ is an independent copy of $(X,Y)$. This raises natural questions about independence of variables like $X-X^{\prime }$ and $Y-Y^{\prime }$, about the connection between Cov$(|X-X^{\prime }|,|Y-Y^{\prime }|)$ 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.

References in corpus (1)

• Measuring and testing dependence by correlation of distances