Consistency of permutation tests of independence using distance covariance, HSIC and dHSIC

Abstract

The Hilbert–Schmidt independence criterion (HSIC) and its d‐variable extension dHSIC are measures of (joint) dependence between random variables. While combining these statistics with a permutation test has become a popular method of testing the null hypothesis of (joint) independence, it had thus far not been proved that this results in a consistent test. In this work, we provide a simple proof that the permutation test with the test statistic HSIC or dHSIC is indeed consistent when using characteristic kernels. That is, we prove that under each alternative hypothesis, the power of these permutation tests indeed converges to 1 as the sample size converges to infinity. Since the test is consistent for each number of permutations, we further give a brief discussion of how the number of permutations relates to the power of the test and how the number of permutations may be selected in practice.