Appears in collection : 2022 - T3 - WS3 - Measure-theoretic Approaches and Optimal Transportation in Statistics

In this work, we consider the question of the optimality of two-sample tests based on MMD. Based on an operator perspective, we propose a spectral regularized two-sample test involving both the mean element and covariance operator and show it to be minimax optimal for a suitable class of alternatives, while the MMD test fails to be optimal. Since the proposed test requires knowledge of the regularization parameter, we propose an adaptive test based on aggregation and show it to be minimax optimal up to logarithmic factors.