Minimax Goodness-of-Fit Testing in Wasserstein Distance

Music City Center 

The empirical Wasserstein distance has been advocated as a natural test statistic for multivariate goodness-of-fit testing, and has been the subject of intensive study in the recent literature on statistical optimal transport. This body of work has characterized the limiting distribution of the empirical Wasserstein distance (and its regularized counterparts) in increasing levels of generality, which enables the construction of asymptotically valid critical values. Despite these methodological advances, theory has lagged behind on characterizing the power of this and related tests against alternatives separated from the null hypothesis in Wasserstein distance, which is typically the intended set of alternatives. This talk will provide some new steps in this direction. We adopt the minimax perspective, and derive the critical radius for goodness-of-fit testing under the Wasserstein distance, subject to various structural assumptions on the set of alternatives. We derive several simple and intuitive tests which are minimax optimal, and we present some surprises regarding the (sub)optimality of various commonly-used test statistics. This talk is based on joint work with Sivaraman Balakrishnan.

Optimal transport