Asymptotic distribution‐free tests related to maximum mean discrepancy

The topic of maximum mean discrepancy has been prominent in statistical analyses of multivariate two‐sample goodness of fit. Despite its usefulness, one major bottleneck is the testing process: Because the null distribution of maximum mean discrepancy depends on the underlying distribution, it typically requires a permutation test to estimate the null and compute the p$$ p $$‐value, which is very costly for a large amount of data. To overcome the difficulty, in this article, we propose combined probability tests based on the equality of the means of a characteristic kernel, which arises in maximum mean discrepancy analysis of multivariate data. The combined probability tests are shown to be asymptotically distribution‐free and therefore have well‐known critical values. We further show that the proposed tests are consistent against all fixed alternatives under the existence of the second moment of the characteristic kernel. A local power analysis provides strong support for the new approach by establishing the nontrivial power of our tests over square root‐(m+n)−1$$ {\left(m+n\right)}^{-1} $$ neighborhoods. We illustrate the advantages of the proposed method via simulation studies and a gene expression dataset analysis.