The American Statistician

Distance covariance (Székely et al., 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.