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We provide a unifying framework linking two classes of statistics used intwo-sample and independence testing: on the one hand, the energy distances anddistance covariances from the statistics literature; on the other, maximum meandiscrepancies (MMD), that is, distances between embeddings of distributions toreproducing kernel Hilbert spaces (RKHS), as established in machine learning.In the case where the energy distance is computed with a semimetric of negativetype, a positive definite kernel, termed distance kernel, may be defined suchthat the MMD corresponds exactly to the energy distance. Conversely, for anypositive definite kernel, we can interpret the MMD as energy distance withrespect to some negative-type semimetric. This equivalence readily extends todistance covariance using kernels on the product space. We determine the classof probability distributions for which the test statistics are consistentagainst all alternatives. Finally, we investigate the performance of the familyof distance kernels in two-sample and independence tests: we show in particularthat the energy distance most commonly employed in statistics is just onemember of a parametric family of kernels, and that other choices from thisfamily can yield more powerful tests.

Equivalence of distance-based and RKHS-based statistics in hypothesis testing