Exponentially Consistent Kernel Two-Sample Tests
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Given two sets of independent samples from unknown distributions P and Q, a two-sample test decides whether to reject the null hypothesis that P=Q. Recent attention has focused on kernel two-sample tests as the test statistics are easy to compute, converge fast, and have low bias with their finite sample estimates. However, there still lacks an exact characterization on the asymptotic performance of such tests, and in particular, the rate at which the type-II error probability decays to zero in the large sample limit. In this work, we show that a class of kernel two-sample tests are exponentially consistent on Polish, locally compact Hausdorff space, e.g., R^d. The obtained exponential decay rate is further shown to be optimal among all two-sample tests meeting the given level constraint, and is independent of particular kernels provided that they are bounded continuous and characteristic. Key to our approach are an extended version of Sanov's theorem and a recent result that identifies the Maximum Mean Discrepancy as a metric of weak convergence of probability measures.
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