The Annals of Statistics
- Ann. Statist.
- Volume 41, Number 5 (2013), 2263-2291.
Equivalence of distance-based and RKHS-based statistics in hypothesis testing
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.
Ann. Statist., Volume 41, Number 5 (2013), 2263-2291.
First available in Project Euclid: 5 November 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62G10: Hypothesis testing 62H20: Measures of association (correlation, canonical correlation, etc.) 68Q32: Computational learning theory [See also 68T05]
Secondary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
Sejdinovic, Dino; Sriperumbudur, Bharath; Gretton, Arthur; Fukumizu, Kenji. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Ann. Statist. 41 (2013), no. 5, 2263--2291. doi:10.1214/13-AOS1140. https://projecteuclid.org/euclid.aos/1383661264