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 techniques 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 satisfied. 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 the power performance of the different tests depends on the specific alternative hypothesis.