= Gaussian diagonal-covariance likelihood-ratio test
{c}
{title2=$\Lambda=|R|^{n/2}$}
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 $D=\operatorname{diag}(S_{11},\ldots,S_{pp})$. Both fitted quadratic terms equal $np$, so their likelihood quotient is $(|S|/|D|)^{n/2}=|R|^{n/2}$, where $R=D^{-1/2}SD^{-1/2}$ is the <sample correlation matrix>. Small determinants give evidence against diagonality. For two variables this is a two-sided test of their <sample correlation>.
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