  For a multivariate linear model, Wilk's likelihood ratio test (LRT)
constitutes one of the cornerstone tools. However, the computation of its
quantiles under the null or the alternative requires complex analytic
approximations and more importantly, these distributional approximations are
feasible only for moderate dimension of the dependent variable, say $p\le 20$.
On the other hand, assuming that the data dimension $p$ as well as the number
$q$ of regression variables are fixed while the sample size $n$ grows, several
asymptotic approximations are proposed in the literature for Wilk's $\bLa$
including the widely used chi-square approximation. In this paper, we consider
necessary modifications to Wilk's test in a high-dimensional context,
specifically assuming a high data dimension $p$ and a large sample size $n$.
Based on recent random matrix theory, the correction we propose to Wilk's test
is asymptotically Gaussian under the null and simulations demonstrate that the
corrected LRT has very satisfactory size and power, surely in the large $p$ and
large $n$ context, but also for moderately large data dimensions like $p=30$ or
$p=50$. As a byproduct, we give a reason explaining why the standard chi-square
approximation fails for high-dimensional data. We also introduce a new
procedure for the classical multiple sample significance test in MANOVA which
is valid for high-dimensional data.
