Comparing a large number of multivariate distributions

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