A Simple Two-Sample Test in High Dimensions Based on L²-Norm

Testing the equality of two means is a fundamental inference problem. For high-dimensional data, the Hotelling’s T²-test either performs poorly or becomes inapplicable. Several modifications have been proposed to address this issue. However, most of them are based on asymptotic normality of the null distributions of their test statistics which inevitably requires strong assumptions on the covariance. We study this problem thoroughly and propose an L²-norm based test that works under mild conditions and even when there are fewer observations than the dimension. Specially, to cope with general nonnormality of the null distribution we employ the Welch–Satterthwaite χ²-approximation. We derive a sharp upper bound on the approximation error and use it to justify that χ²-approximation is preferred to normal approximation. Simple ratio-consistent estimators for the parameters in the χ²-approximation are given. Importantly, our test can cope with singularity or near singularity of the covariance which is commonly seen in high dimensions and is the main cause of nonnormality. The power of the proposed test is also investigated. Extensive simulation studies and an application show that our test is at least comparable to and often outperforms several competitors in terms of size control, and the powers are comparable when their sizes are. Supplementary materials for this article are available online.