Distance Covariance, Independence, and Pairwise Differences

Jakob Raymaekers and Peter Rousseeuw

The American Statistician, 2025, vol. 79, issue 1, 122-128

Abstract: 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 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.

Date: 2025
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DOI: 10.1080/00031305.2024.2374966

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