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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, distancesbetween embeddings of distributions to reproducing kernel Hilbert spaces(RKHS), as established in machine learning. The equivalence holds when energydistances are computed with semimetrics of negative type, in which case akernel may be defined such that the RKHS distance between distributionscorresponds exactly to the energy distance. We determine the class ofprobability distributions for which kernels induced by semimetrics arecharacteristic (that is, for which embeddings of the distributions to an RKHSare injective). Finally, we investigate the performance of this family ofkernels in two-sample and independence tests: we show in particular that theenergy distance most commonly employed in statistics is just one member of aparametric family of kernels, and that other choices from this family can yieldmore powerful tests.

Hypothesis testing using pairwise distances and associated kernels (with Appendix)