A Basic Treatment of the Distance Covariance

TL;DR

This paper provides a rigorous, introductory explanation of distance covariance, a measure of dependence between multivariate variables that detects all types of associations, suitable for educational purposes.

Contribution

It offers a foundational, accessible treatment of distance covariance, facilitating its teaching at undergraduate and early graduate levels.

Findings

Distance covariance equals zero if and only if variables are independent.

It effectively detects all types of non-linear associations.

Provides a basis for educational presentation of the concept.

Abstract

The distance covariance of Sz\'ekely, et al. [23] and Sz\'ekely and Rizzo [21], a powerful measure of dependence between sets of multivariate random variables, has the crucial feature that it equals zero if and only if the sets are mutually independent. Hence the distance covariance can be applied to multivariate data to detect arbitrary types of non-linear associations between sets of variables. We provide in this article a basic, albeit rigorous, introductory treatment of the distance covariance. Our investigations yield an approach that can be used as the foundation for presentation of this important and timely topic even in advanced undergraduate- or junior graduate-level courses on mathematical statistics.