Directional testing for one-way MANOVA in divergent dimensions

Testing the equality of mean vectors across $g$ different groups plays an important role in many scientific fields. In regular frameworks, likelihood-based statistics under the normality assumption offer a general solution to this task. However, the accuracy of standard asymptotic results is not reliable when the dimension $p$ of the data is large relative to the sample size $n_i$ of each group. We propose here an exact directional test for the equality of $g$ normal mean vectors with identical unknown covariance matrix in a high dimensional setting, provided that $\sum_{i=1}^g n_i \ge p+g+1$. In the case of two groups ($g=2$), the directional test coincides with the Hotelling's $T^2$ test. In the more general situation where the $g$ independent groups may have different unknown covariance matrices, although exactness does not hold, simulation studies show that the directional test is more accurate than most commonly used likelihood{-}based solutions, at least in a moderate dimensional setting in which $p=O(n_i^\tau)$, $\tau \in (0,1)$. Robustness of the directional approach and its competitors under deviation from the assumption of multivariate normality is also numerically investigated. Our proposal is here applied to data on blood characteristics of male athletes and to microarray data storing gene expressions in patients with breast tumors.