Game-theoretic Formulations of Sequential Nonparametric One- and Two-Sample Tests

We study the problem of designing consistent sequential one- and two-sample tests in a nonparametric setting. Guided by the principle of \emph{testing by betting}, we reframe the task of constructing sequential tests into that of selecting payoff functions that maximize the wealth of a fictitious bettor, betting against the null in a repeated game. The resulting sequential test rejects the null when the bettor's wealth process exceeds an appropriate threshold. We propose a general strategy for selecting payoff functions as predictable estimates of the \emph{witness function} associated with the variational representation of some statistical distance measures, such as integral probability metrics~(IPMs) and $\varphi$-divergences. Overall, this approach ensures that (i) the wealth process is a non-negative martingale under the null, thus allowing tight control over the type-I error, and (ii) it grows to infinity almost surely under the alternative, thus implying consistency. We accomplish this by designing composite e-processes that remain bounded in expectation under the null, but grow to infinity under the alternative. We instantiate the general test for some common distance metrics to obtain sequential versions of Kolmogorov-Smirnov~(KS) test, $\chi^2$-test and kernel-MMD test, and empirically demonstrate their ability to adapt to the unknown hardness of the problem. The sequential testing framework constructed in this paper is versatile, and we end with a discussion on applying these ideas to two related problems: testing for higher-order stochastic dominance, and testing for symmetry.