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Write $v={\sigma }^2$. For one observation, the score isandUnder the restriction $v=1$, the restricted maximum-likelihood estimator is $\tilde{\mu }=\bar{X}$. At $(\bar{X},1)$ the first score component is zero, while the second isSince $I_1(\bar{X},1{)}^{-1}=\mathrm{diag}(1,2)$, substitution into the score statistic gives

Restricted maximum likelihood estimates covariance matrix parameters from linear combinations of the data whose distribution does not involve the fixed mean coefficients. It differs from a restricted maximum-likelihood estimator obtained by maximizing an ordinary likelihood function under a null hypothesis.