Testing the equality of two multivariate distributions is critical to ensuring valid and reasonable statistical conclusions. Since the true alternative is unknown in practice, developing a powerful testing strategy remains an open question. We propose a composite kernel-based test that flexibly assigns weights to candidate tests, thereby effectively enhancing power. Our test is robust and adept at handling various alternative hypotheses. Furthermore, our test is data-adaptive and sensitive to detecting distributional differences across both local and global features. The asymptotic distributions are investigated under both the null and the alternative hypotheses. In addition, we present the asymptotic behavior of the test statistic under a range of Pitman local alternative hypotheses. To improve the finite sample properties of the proposed test, we employ a permutation method to obtain critical values or p-values. We also show the theoretical validity of the permutation method. Compared with current state-of-the-art test methods, the proposed test exhibits satisfactory performance, as evidenced by extensive simulations and real-world data.
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