In this paper, we propose a test for the equality of multiple distributions based on kernel mean embeddings. Our framework provides a flexible way to handle multivariate data by virtue of kernel methods and allows the number of distributions to increase with the sample size. This is in contrast to previous studies that have been mostly restricted to classical univariate settings with a fixed number of distributions. By building on Cramér-type moderate deviation for degenerate two-sample $V$-statistics, we derive the limiting null distribution of the test statistic and show that it converges to a Gumbel distribution. The limiting distribution, however, depends on an infinite number of nuisance parameters, which makes it infeasible for use in practice. To address this issue, the proposed test is implemented via the permutation procedure and is shown to be minimax rate optimal against sparse alternatives. During our analysis, an exponential concentration inequality for the permuted test statistic is developed which may be of independent interest.
"Comparing a large number of multivariate distributions." Bernoulli 27 (1) 419 - 441, February 2021. https://doi.org/10.3150/20-BEJ1244