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We characterize the asymptotic performance of nonparametric one- andtwo-sample testing. The exponential decay rate or error exponent of the type-IIerror probability is used as the asymptotic performance metric, and an optimaltest achieves the maximum rate subject to a constant level constraint on thetype-I error probability. With Sanov's theorem, we derive a sufficientcondition for one-sample tests to achieve the optimal error exponent in theuniversal setting, i.e., for any distribution defining the alternativehypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD)based tests attain the optimal type-II error exponent on $\mathbb R^d$, whilethe quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve thisoptimality with an asymptotic level constraint. For general two-sample testing,however, Sanov's theorem is insufficient to obtain a similar sufficientcondition. We proceed to establish an extended version of Sanov's theorem andderive an exact error exponent for the quadratic-time MMD based two-sampletests. The obtained error exponent is further shown to be optimal among alltwo-sample tests satisfying a given level constraint. Our work hence providesan achievability result for optimal nonparametric one- and two-sample testingin the universal setting. Application to off-line change detection and relatedissues are also discussed.

Asymptotically Optimal One- and Two-Sample Testing with Kernels