Open Access
December 2009 Brownian distance covariance
Gábor J. Székely, Maria L. Rizzo
Ann. Appl. Stat. 3(4): 1236-1265 (December 2009). DOI: 10.1214/09-AOAS312


Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. Distance correlation characterizes independence: it is zero if and only if the random vectors are independent. The notion of covariance with respect to a stochastic process is introduced, and it is shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product-moment covariance, as we obtain Pearson product-moment covariance by replacing the Brownian motion in the definition with identity. The corresponding statistic has an elegantly simple computing formula. Advantages of applying Brownian covariance and correlation vs the classical Pearson covariance and correlation are discussed and illustrated.


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Gábor J. Székely. Maria L. Rizzo. "Brownian distance covariance." Ann. Appl. Stat. 3 (4) 1236 - 1265, December 2009.


Published: December 2009
First available in Project Euclid: 1 March 2010

zbMATH: 1196.62077
MathSciNet: MR2752127
Digital Object Identifier: 10.1214/09-AOAS312

Keywords: Brownian covariance , dcor , Distance correlation , independence , multivariate

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.3 • No. 4 • December 2009
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