Robust Multivariate Nonparametric Tests via Projection-Pursuit
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In this work, we generalize the Cramér-von Mises statistic via projection pursuit to obtain robust tests for the multivariate two-sample problem. The proposed tests are consistent against all fixed alternatives, robust to heavy-tailed data and minimax rate optimal. Our test statistics are completely free of tuning parameters and are computationally efficient even in high dimensions. When the dimension tends to infinity, the proposed test is shown to have identical power to that of the existing high-dimensional 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 to the reproducing kernel Hilbert space approach. In addition to the Cramér-von Mises statistic, we show that the projection pursuit technique can be used to define robust, multivariate tests in many other problems.
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