Distance multivariance: New dependence measures for random vectors

TL;DR

This paper introduces distance multivariance and total distance multivariance, new dependence measures for multiple random variables based on characteristic functions, extending the concept of distance covariance to higher dimensions.

Contribution

The paper presents novel dependence measures for multiple variables, extending distance covariance, with practical finite-sample representations and consistent independence testing.

Findings

Total distance multivariance detects independence of multiple variables.

Finite-sample representation uses distance matrices and negative definite functions.

Provides a consistent test for independence under mild conditions.

Abstract

We introduce two new measures for the dependence of $n \ge 2$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted $L^2$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Sz\'ekely, Rizzo and Bakirov) from pairs of random variables to $n$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of $n$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.