Test for mean matrix in GMANOVA model under heteroscedasticity and non-normality for high-dimensional data

Abstract

This paper is concerned with the testing bilateral linear hypothesis on the mean matrix in the context of the generalized multivariate analysis of variance (GMANOVA) model when the dimensions of the observed vector may exceed the sample size, the design may become unbalanced, the population may not be normal, or the true covariance matrices may be unequal. The suggested testing methodology can treat many problems such as the one- and two-way MANOVA tests, the test for parallelism in profile analysis, etc., as specific ones. We propose a bias-corrected estimator of the Frobenius norm for the mean matrix, which is a key component of the test statistic. The null and non-null distributions are derived under a general high-dimensional asymptotic framework that allows the dimensionality to arbitrarily exceed the sample size of a group, thereby establishing consistency for the testing criterion. The accuracy of the proposed test in a finite sample is investigated through simulations conducted for several high-dimensional scenarios and various underlying population distributions in combination with different within-group covariance structures. Finally, the proposed test is applied to a high-dimensional two-way MANOVA problem for DNA microarray data.