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October 2013 Equivalence of distance-based and RKHS-based statistics in hypothesis testing
Dino Sejdinovic, Bharath Sriperumbudur, Arthur Gretton, Kenji Fukumizu
Ann. Statist. 41(5): 2263-2291 (October 2013). DOI: 10.1214/13-AOS1140

Abstract

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. In the case where the energy distance is computed with a semimetric of negative type, a positive definite kernel, termed distance kernel, may be defined such that the MMD corresponds exactly to the energy distance. Conversely, for any positive definite kernel, we can interpret the MMD as energy distance with respect to some negative-type semimetric. This equivalence readily extends to distance covariance using kernels on the product space. We determine the class of probability distributions for which the test statistics are consistent against all alternatives. Finally, we investigate the performance of the family of distance kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.

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Dino Sejdinovic. Bharath Sriperumbudur. Arthur Gretton. Kenji Fukumizu. "Equivalence of distance-based and RKHS-based statistics in hypothesis testing." Ann. Statist. 41 (5) 2263 - 2291, October 2013. https://doi.org/10.1214/13-AOS1140

Information

Published: October 2013
First available in Project Euclid: 5 November 2013

zbMATH: 1281.62117
MathSciNet: MR3127866
Digital Object Identifier: 10.1214/13-AOS1140

Subjects:
Primary: 62G10, 62H20, 68Q32
Secondary: 46E22

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 5 • October 2013
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