 Solution

Wilks theorem states that, under an interior null parameter and the usual identifiability, smoothness and nonsingular-information regularity conditions, $-2\log \Lambda$ for nested models converges under the null to a chi-square distribution with degrees of freedom equal to the difference of parameter dimensions. Boundary or nonidentifiable nulls need not have this distribution.

Write $\bar{X}=n^{-1}\sum X_i$, $Q=\sum (X_i-\bar{X}{)}^2$ and $S=\sum X_i^2=Q+n{\bar{X}}^2$. The unrestricted maximum-likelihood estimates are $\hat{\mu }=\bar{X}$ and ${\hat{\sigma }}^2=Q/n$. Under the null the variance estimate is $S/n$. Substituting both maxima into the Gaussian likelihood givesThis is for $n\ge 2$, with positive true variance; $Q>0$ almost surely. The lost parameter dimension is one, even though variance is estimated in both models.

The two-sided Student t-test uses $T=\sqrt{n}\bar{X}/s$, where $s^2=Q/(n-1)$, and rejects when $|T|>t_{n-1,1-\alpha /2}$. Under the null $T$ has exactly the Student t-distribution with $n-1$ degrees of freedom. The likelihood-ratio statistic isunder the null, so its chi-square approximation agrees asymptotically with the squared asymptotic standard-normal t statistic. In fact the expression is strictly increasing in $T^2$: with exact finite-sample calibration the tests have identical rejection regions for every $n$, not just approximately for large samples.