  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.
