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.

中文翻译：

---

使用距离协方差，HSIC和dHSIC的独立性置换检验的一致性

希尔伯特-施密特独立性标准（HSIC）及其d变量扩展dHSIC是对随机变量之间（联合）依赖性的度量。尽管将这些统计数据与置换检验相结合已成为检验（联合）独立性零假设的一种流行方法，但迄今为止尚未证明这会导致一致性检验。在这项工作中，我们提供了一个简单的证明，即在使用特征核时，带有检验统计量HSIC或dHSIC的置换检验确实是一致的。也就是说，我们证明了在每个备选假设下，随着样本量收敛至无穷大，这些置换检验的功效确实收敛于1。由于测试对于每个排列数都是一致的，因此我们进一步简要讨论排列数与测试功效之间的关系以及在实践中如何选择排列数。