 Solution

With the divisor-$n$ sample covariance matrix defined above, the maximum-likelihood estimators areTheir joint sampling distribution is specified byHere $W_p(\nu ,V)$ uses the Wishart distribution convention $\mathbb{E}W=\nu V$. Thus the covariance maximum-likelihood estimator has expected value $(n-1)V/n$; the unbiased estimator of the covariance matrix is $nS/(n-1)$. These distributions follow from orthogonal projections of the jointly multivariate normal observations onto the constant-observation direction and its orthogonal complement, though no proof is required here. The covariance maximum in the positive-definite parameter space requires $S$ nonsingular, which holds almost surely when $n>p$ and $V$ is positive definite. Otherwise the rank of a centered sample covariance matrix shows why the usual unrestricted finite maximum does not exist.