Testing linear hypotheses of mean vectors for high-dimension data with unequal covariance matrices

Abstract

We propose a new test procedure for testing linear hypothesis on the mean vectors of normal populations with unequal covariance matrices when the dimensionality, p exceeds the sample size N, i.e. p/N→c<∞. Our procedure is based on the Dempster trace criterion and is shown to be consistent in high dimensions.The asymptotic null and non-null distributions of the proposed test statistic are established in the high dimensional setting and improved estimator of the critical point of the test is derived using Cornish–Fisher expansion. As a special case, our testing procedure is applied to multivariate Behrens–Fisher problem. We illustrate the relevance and benefits of the proposed approach via Monte-Carlo simulations which show that our new test is comparable to, and in many cases is more powerful than, the tests for equality of means presented in the recent literature.