Generalizing Distance Covariance to Measure and Test Multivariate Mutual Dependence

TL;DR

This paper introduces three new measures of mutual dependence among multiple random vectors, extending distance covariance to capture complex non-linear dependencies and providing practical tests with demonstrated effectiveness.

Contribution

It generalizes distance covariance to multivariate mutual dependence and develops empirical tests with theoretical properties and practical validation.

Findings

Measures are zero iff vectors are mutually independent

Proposed tests effectively detect complex dependencies

Empirical results validate the measures and tests

Abstract

We propose three measures of mutual dependence between multiple random vectors. All the measures are zero if and only if the random vectors are mutually independent. The first measure generalizes distance covariance from pairwise dependence to mutual dependence, while the other two measures are sums of squared distance covariance. All the measures share similar properties and asymptotic distributions to distance covariance, and capture non-linear and non-monotone mutual dependence between the random vectors. Inspired by complete and incomplete V-statistics, we define the empirical measures and simplified empirical measures as a trade-off between the complexity and power when testing mutual independence. Implementation of the tests is demonstrated by both simulation results and real data examples.