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arXiv · 1711.07775

Distance multivariance: New dependence measures for random vectors

Björn Böttcher·Martin Keller-Ressel·René L. Schilling
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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 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.

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Björn Böttcher, Martin Keller-Ressel, René L. Schilling. 2018-10-17. Distance multivariance: New dependence measures for random vectors. https://doi.org/10.1214/18-aos1764

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