Distance multivariance: New dependence measures for random vectors

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.

arXiv commenttitle changed; completely restructured; new content: comparison with dHSIC and Example 5.2; accepted for publication in AoS

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• Rejoinder: Brownian distance covariance
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Cited by in corpus (8)

• Detecting independence of random vectors: generalized distance covariance and Gaussian covariance
• Nonparametric Statistical Inference via Metric Distribution Function in Metric Spaces
• On complex Gaussian random fields, Gaussian quadratic forms and sample distance multivariance
• Copula versions of distance multivariance and dHSIC via the distributional transform -- a general approach to construct invariant dependence measures
• Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin
• Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance
• Notes on the interpretation of dependence measures
• A nonparametric distribution-free test of independence among continuous random vectors based on \texorpdfstring{$L_1$}{}-norm