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Under the unrestricted model, insert the maximum-likelihood estimates $(\bar{x},S)$. Under the diagonal-covariance restriction, the quadratic form is minimized over the mean at $\bar{x}$, and the remaining objective separates over the diagonal entries:Differentiating in $v_j>0$ gives ${\hat{v}}_j=S_{jj}$; the derivative changes from negative to positive there. Put $D=\mathrm{diag}(S_{11},\dots ,S_{pp})$. The fitted trace term is $p$ under either model, since $\mathrm{tr}(D^{-1}S)=p=\mathrm{tr}(S^{-1}S)$. Consequently the likelihood-ratio test statistic is determined byThis is the Gaussian diagonal-covariance likelihood-ratio test. Reject for sufficiently small $\log |R|$, equivalently for large $-n\log |R|$. Under the regular large-sample calibration, Wilks theorem gives $-n\log |R|\Rightarrow {\chi }_{p(p-1)/2}^2$, since diagonality removes that many free covariance parameters. The exact critical constant can instead be obtained from the null distribution at the actual sample size.

For $p=2$, write the sample correlation coefficient as $r$. Then $|R|=1-r^2$, so the likelihood-ratio test rejects for large $|r|$, not merely for large positive $r$. Under the null, the two variables have a bivariate normal distribution and are independent, and the exact test isIndeed, condition on the centered first-variable observation vector. Decompose the second centered Gaussian vector into its component along that direction and the $n-2$-dimensional orthogonal complement. Their squared lengths give independent ${\chi }_1^2$ and ${\chi }_{n-2}^2$ variables. Their ratio shows that $r\sqrt{(n-2)/(1-r^2)}$ has Student's t-distribution with $n-2$ statistical degrees of freedom.