Directional testing for high dimensional multivariate normal distributions

Thanks to its favorable properties, the multivariate normal distribution is still largely employed for modeling phenomena in various scientific fields. However, when the number of components p is of the same asymptotic order as the sample size n, standard inferential tech- niques are generally inadequate to conduct hypothesis testing on the mean vector and/or the covariance matrix. Within several prominent frameworks, we propose then to draw reliable conclusions via a directional test. We show that under the null hypothesis the directional p- value is exactly uniformly distributed even when p is of the same order of n, provided that conditions for the existence of the maximum likelihood estimate for the normal model are satis- fied. Extensive simulation results confirm the theoretical findings across different values of p/n, and show that under the null hypothesis the directional test outperforms not only the usual first and higher-order finite-p solutions but also alternative methods tailored for high-dimensional settings. Simulation results also indicate that power performance of the different tests depends on the specific alternative hypothesis.