## The Annals of Statistics

### Partial distance correlation with methods for dissimilarities

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 42, Number 6 (2014), 2382-2412.

Dates
First available in Project Euclid: 20 October 2014

https://projecteuclid.org/euclid.aos/1413810731

Digital Object Identifier
doi:10.1214/14-AOS1255

Mathematical Reviews number (MathSciNet)
MR3269983

Zentralblatt MATH identifier
1309.62105

#### Citation

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

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