Distance multivariance: New dependence measures for random vectors

Abstract

We introduce two new measures for the dependence of n ≥ 2 random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted L2-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, 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.