Hypothesis testing using pairwise distances and associated kernels (with Appendix)
This work provides a theoretical bridge between statistical and machine learning methods for hypothesis testing, potentially improving test power for researchers in these fields, though it is incremental in nature.
The paper establishes a unifying framework linking energy distances and distance covariances from statistics with RKHS distances from machine learning, showing equivalence under semimetrics of negative type and identifying conditions for characteristic kernels. It demonstrates that the commonly used energy distance is part of a parametric family, and other choices from this family can yield more powerful two-sample and independence tests.
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, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. The equivalence holds when energy distances are computed with semimetrics of negative type, in which case a kernel may be defined such that the RKHS distance between distributions corresponds exactly to the energy distance. We determine the class of probability distributions for which kernels induced by semimetrics are characteristic (that is, for which embeddings of the distributions to an RKHS are injective). Finally, we investigate the performance of this family of 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.