 Gaussian diagonal-covariance likelihood-ratio test

For independent multivariate normal observations with unrestricted mean and positive-definite sample covariance matrix $S=n^{-1}{\sum }_i(x_i-\bar{x})(x_i-\bar{x}{)}^T$, the unrestricted maximum-likelihood estimates are $\bar{x},S$. Requiring a diagonal population covariance matrix gives the same mean and covariance estimate $D=\mathrm{diag}(S_{11},\dots ,S_{pp})$. Both fitted quadratic terms equal $np$, so their likelihood quotient is $(|S|/|D|{)}^{n/2}=|R{|}^{n/2}$, where $R=D^{-1/2}S{D}^{-1/2}$ is the sample correlation matrix. Small determinants give evidence against diagonality. For two variables this is a two-sided test of their sample correlation.