On two-sample mean tests under spiked covariances

Abstract

This paper considers testing the means of two p-variate normal samples in high dimensional settings. We show that under the null hypothesis, a necessary and sufficient condition for the asymptotic normality of the test statistic of Chen and Qin (2010) is that the eigenvalues of the covariance matrix are concentrated around their average. However, this condition is not satisfied when the variables are strongly correlated. To characterize the correlations between variables, we adopt a spiked covariance model. Under the spiked covariance model, we derive the asymptotic distribution of the test statistic of Chen and Qin (2010) and correct its critical value. The recently proposed random projection test procedures suggest that the power of tests may be boosted using the projected data. By maximizing an average signal to noise ratio, we find that the optimal projection subspace is the orthogonal complement of the principal subspace. We propose a new test procedure through the projection onto the estimated orthogonal complement of the principal subspace. The asymptotic normality of the new test statistic is proved and the asymptotic power function of the test is given. Theoretical and simulation results show that the new test outperforms the competing tests substantially under the spiked covariance model.