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ROBUST MULTIVARIATE NONPARAMETRIC TESTS VIA PROJECTION AVERAGING
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DOI：10.1214/19-AOS1936.png)
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Abstract
En 中文
In this work, we generalize the Cramer-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 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 method to the reproducing kernel Hilbert space approach. In addition to the Cramer-von Mises statistic, we demonstrate that the projection-averaging technique can be used to define robust multivariate tests in many other problems.
Keywords:
Energy statistic
high dimension and low sample size
independence testing
maximum mean discrepancy
permutation tests
U-statistic