Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 43 / 2 / i / Solution

Use the usual sample mean and the divisor-$n-1$ sample covariance matrix,The residual sum is zero, so expanding $x_j-\mu =(x_j-\bar{x})+(\bar{x}-\mu )$ givesMultiplying the independent multivariate normal densities and taking logarithms yields the log-likelihoodHere $|V|$ is the determinant and $\mathrm{tr}$ denotes the trace. The maximum-likelihood estimators areIf $S$ instead denotes the divisor-$n$ sample covariance matrix, replace $(n-1)S$ by $nS$ and then $\hat{V}=S$. These formulas describe a nonsingular maximum-likelihood estimator when the empirical scatter is positive-definite; if it is singular, there is no maximizer over positive-definite covariance matrices.