The Annals of Statistics
- Ann. Statist.
- Volume 42, Number 6 (2014), 2382-2412.
Partial distance correlation with methods for dissimilarities
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 Hilbert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hilbert 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.
Ann. Statist., Volume 42, Number 6 (2014), 2382-2412.
First available in Project Euclid: 20 October 2014
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62Hxx: Multivariate analysis [See also 60Exx] 62H20: Measures of association (correlation, canonical correlation, etc.) 62H15: Hypothesis testing
Secondary: 62Gxx: Nonparametric inference
Székely, Gábor J.; Rizzo, Maria L. Partial distance correlation with methods for dissimilarities. Ann. Statist. 42 (2014), no. 6, 2382--2412. doi:10.1214/14-AOS1255. https://projecteuclid.org/euclid.aos/1413810731