PARTIAL DISTANCE CORRELATION WITH METHODS FOR DISSIMILARITIES

• Published in: The Annals of statistics 2014-12, Vol.42 (6), p.2382-2412
• Main Authors: Székely, Gábor J., Rizzo, Maria L.
• Format: Article
• Language: English
• Subjects: 62Gxx; 62H15; 62H20; 62Hxx; Correlation analysis; Correlations; Covariance; Covariance matrices; Definitions; dissimilarity; energy statistics; Estimating techniques; Euclidean space; False positive errors; Hilbert space; Hilbert spaces; Independence; Inner products; Mantels; Mathematical problems; Mathematical vectors; multivariate; partial distance correlation; Random sampling; Studies; Vector space
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/14-AOS1255

Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions and applications of distance correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation, we introduce a new Hubert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hubert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.