Tests for multivariate analysis of variance in high dimension under non-normality

In this article, we consider the problem of testing the equality of mean vectors of dimension pp of several groups with a common unknown non-singular covariance matrix Σ, based on NN independent observation vectors where NN may be less than the dimension pp. This problem, known in the literature as the multivariate analysis of variance (MANOVA) in high-dimension has recently been considered in the statistical literature by Srivastava and Fujikoshi (2006) [8], Srivastava (2007) [5] and Schott (2007) [3]. All these tests are not invariant under the change of units of measurements. On the lines of Srivastava and Du (2008) [7] and Srivastava (2009) [6], we propose a test that has the above invariance property. The null and the non-null distributions are derived under the assumption that (N,p)→∞(N,p)→∞ and NN may be less than pp and the observation vectors follow a general non-normal model.