Gaussian covariance diagonality likelihood-ratio test ($W=n\log ({\prod }_j{S}_{jj}/\det S)$)

For independent observations from a nonsingular multivariate normal distribution with unknown mean, let $S=n^{-1}{\sum }_i(y_i-\bar{y})(y_i-\bar{y}{)}^T$. The unrestricted maximum-likelihood estimator of covariance is $S$, while the diagonal restriction gives $D=\mathrm{diag}(S_{11},\dots ,S_{pp})$. The fitted trace terms both equal $p$, so twice the log-likelihood difference is $W=n\log (\det D/\det S)=-n\log \det R$, where $R$ is the sample correlation matrix. Under the diagonal null hypothesis and fixed dimension, the Wilks theorem gives $W\Rightarrow {\chi }_{p(p-1)/2}^2$.