A Simple Scale-Invariant Two-Sample Test for High-dimensional Data

Abstract

A new scale-invariant test for two-sample problems for high-dimensional data is proposed and studied. Under some regularity conditions and the null hypothesis, the proposed test statistic and a chi-square-type mixture are shown to have the same limiting distribution after they are normalized. The limiting distribution can be normal or non-normal, depending on the underlying covariance structure of the high-dimensional data. To approximate the null distribution of the proposed test, the well-known Welch-Satterthwaite chi-square approximation is applied. The resulting test is shown to be adaptive to the shape of the underlying null distribution in the sense that when the test statistic is asymptotically normally distributed under the null hypothesis, so is the approximation distribution, and when the approximation distribution is asymptotically non-normally distributed, so is the underlying null distribution of the test statistic. The asymptotic powers of the proposed test under some local alternatives are derived. Simulation studies and a real data application are used to demonstrate the good performance of the proposed test compared with several existing competitors in the literature.