Papers › Distance multivariance: New dependence measures for random vectors
Distance multivariance: New dependence measures for random vectors
Björn Böttcher, Martin Keller-Ressel, René L. Schilling
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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 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.
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