Open Access
October 2019 Distance multivariance: New dependence measures for random vectors
Björn Böttcher, Martin Keller-Ressel, René L. Schilling
Ann. Statist. 47(5): 2757-2789 (October 2019). DOI: 10.1214/18-AOS1764


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.


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Björn Böttcher. Martin Keller-Ressel. René L. Schilling. "Distance multivariance: New dependence measures for random vectors." Ann. Statist. 47 (5) 2757 - 2789, October 2019.


Received: 1 December 2017; Revised: 1 September 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114928
MathSciNet: MR3988772
Digital Object Identifier: 10.1214/18-AOS1764

Primary: 62H20
Secondary: 60E10 , 62G10 , 62G15 , 62G20

Keywords: Characteristic function , dependence measure , Gaussian random field , negative definite function , statistical test of independence , stochastic independence

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 5 • October 2019
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