Measuring and testing dependence by correlation of distances

TL;DR

This paper introduces distance correlation, a new dependence measure that is zero only when variables are independent, with applications in testing independence and demonstrated through empirical results.

Contribution

It proposes a novel dependence measure based on Euclidean distances that generalizes classical correlation and provides a practical test for independence.

Findings

Distance correlation is zero if and only if variables are independent.

The measure has a compact representation similar to classical covariance.

Monte Carlo simulations validate the effectiveness of the independence test.

Abstract

Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented.