Distance multivariance: New dependence measures for random vectors
Björn Böttcher, Martin Keller-Ressel, René L. Schilling
The Annals of Statistics · 2019 · Volume 47 · Issue 5
Abstract
We introduce two new measures for the dependence of $n\ge2$ 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 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.
Keywords
60E10, 62G10, 62G15, 62G20, 62H20, Characteristic function, dependence measure, Gaussian random field, negative definite function, statistical test of independence, stochastic independence
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