Robust multivariate nonparametric tests via projection averaging

In this work, we generalize the Cramér-von Mises statistic via projection averaging to obtain a robust test for the multivariate two-sample problem. The proposed test is consistent against all fixed alternatives, robust to heavytailed data and minimax rate optimal against a certain class of alternatives. Our test statistic is completely free of tuning parameters and is computationally efficient even in high dimensions. When the dimension tends to infinity, the proposed test is shown to have comparable power to the existing highdimensional mean tests under certain location models. As a by-product of our approach, we introduce a new metric called the angular distance which can be thought of as a robust alternative to the Euclidean distance. Using the angular distance, we connect the proposed method to the reproducing kernel Hilbert space approach. In addition to the Cramér-von Mises statistic, we demonstrate that the projection-averaging technique can be used to define robust multivariate tests in many other problems.