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

Abstract

This paper develops a unified testing methodology for high-dimensional generalized multivariate analysis of variance (GMANOVA) models. We derive a test of the bilateral linear hypothesis on the mean matrix in a general scenario where the dimensions of the observed vector may exceed the sample size, design may be unbalanced, the population distribution may be non-normal and the underlying group covariance matrices may be unequal. The suggested methodology is suitable for many inferential problems, such as the one-way MANOVA test and the test for multivariate linear hypothesis on the mean in the polynomial growth curve model. As a key component of our test procedure, we propose a bias-corrected estimator of the Frobenius norm of the mean matrix. We derive null and non-null asymptotic distributions of the test statistic under a general high-dimensional asymptotic framework that allows the dimensionality to arbitrarily exceed the sample size of a group. The accuracy of the proposed test in a finite sample setting is investigated through simulations conducted for several high-dimensional scenarios and various underlying population distributions in combination with different within-group covariance structures. For a practical demonstration we consider a daily Canadian temperature dataset that exhibits group structure, and conclude that the interaction of latitude and longitude has no effect to predict the temperature.