Simple and efficient adaptive two-sample tests for high-dimensional data

In this paper, we first consider testing the equality of two mean vectors in the so-called “high-dimension, low sample size” setting. Different from the existing methods in the literature, we propose a new adaptive test for this two-sample problem based on the order statistics of component-wise two-sample t-statistics. The proposed test is easy to implement and is carried out using permutations, therefore it does not rely on assumptions about normality or the structure of the covariance matrix. Furthermore, our simulation studies show that the proposed test is efficient for detecting different types of differences between two mean vectors, and compares favorably with the existing adaptive tests across a variety of settings. Using the same idea of incorporating the order statistics into the test, we further propose an adaptive two-sample test for testing location differences when marginal distributions have heavy-tails and another adaptive two-sample test for testing arbitrary marginal distribution differences, given high-dimensional, low sample size data. We also demonstrate the usefulness of these tests in high-dimensional, low sample size situations using simulated data as well as real data.