High-dimensional two-sample mean vectors test and support recovery with factor adjustment

Testing the equality of two mean vectors is a classical problem in multivariate analysis. In this article, we consider the test in the high-dimensional setting. Existing tests often assume that the covariance matrix (or its inverse) of the underlying variables is sparse, which is rarely true in social science due to the existence of latent common factors. In the article, we introduce a maximum-type test statistic based on the factor-adjusted data. The factor-adjustment step increases the signal-to-noise ratio and thus results in more powerful test. We obtain the limiting null distribution of the maximum-type test statistic, which is the extreme value distribution of type I. To overcome the well-known slow convergence rate of the test statistic’s distribution to the limiting extreme value distribution, we also propose a multiplier bootstrap method to improve the finite-sample performance. In addition, a multiple testing procedure with false discovery rate (FDR) control is proposed for identifying specific locations that differ significantly between the two groups. Thorough numerical studies are conducted to show the superiority of the test over other state-of-the-art tests. The performance of the test is also assessed through a real stock market dataset.