The Annals of Statistics

Distance multivariance: New dependence measures for random vectors

Björn Böttcher, Martin Keller-Ressel, and René L. Schilling

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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.

Article information

Ann. Statist., Volume 47, Number 5 (2019), 2757-2789.

Received: December 2017
Revised: September 2018
First available in Project Euclid: 3 August 2019

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Mathematical Reviews number (MathSciNet)

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Secondary: 60E10: Characteristic functions; other transforms 62G10: Hypothesis testing 62G15: Tolerance and confidence regions 62G20: Asymptotic properties

Dependence measure stochastic independence negative definite function characteristic function Gaussian random field statistical test of independence


Böttcher, Björn; Keller-Ressel, Martin; Schilling, René L. Distance multivariance: New dependence measures for random vectors. Ann. Statist. 47 (2019), no. 5, 2757--2789. doi:10.1214/18-AOS1764.

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