The Hellinger Correlation

Abstract

In this article, the defining properties of any valid measure of the dependence between two continuous random variables are revisited and complemented with two original ones, shown to imply other usual postulates. While other popular choices are proved to violate some of these requirements, a class of dependence measures satisfying all of them is identified. One particular measure, that we call the Hellinger correlation, appears as a natural choice within that class due to both its theoretical and intuitive appeal. A simple and efficient nonparametric estimator for that quantity is proposed, with its implementation publicly available in the R package HellCor. Synthetic and real-data examples illustrate the descriptive ability of the measure, which can also be used as test statistic for exact independence testing. Supplementary materials for this article are available online.

KEYWORDS:

• Copula
• Dependence
• Hellinger distance
• Measure
• Nearest-neighbor

Acknowledgments

This article includes results produced on the computational cluster Katana at UNSW Sydney.