Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 42 / 1 / i / Solution

Use the divisor-$n$ sample covariance matrix, since this is the normalization that enters the multivariate normal likelihood function:For a covariance matrix that is a positive-definite matrix $V$, the product of the multivariate normal densities givesTo separate the sample mean from the sample covariance matrix, expand $x_j-\mu =(x_j-\bar{x})+(\bar{x}-\mu )$. The cross terms sum to zero because ${\sum }_j(x_j-\bar{x})=0$. Also $u^T{V}^{-1}u=\mathrm{tr}(V^{-1}u{u}^T)$ by the trace identity. ThereforeThe printed expression suppresses the parameter-independent constant $np\log (2\pi )$; dropping this constant has no effect on maximum likelihood estimation or the likelihood-ratio test.