Likelihood-ratio test for a normal variance ($W=q-n-n\log (q/n),Q\sim {\chi }_{n-1}^2$)

For an independent normal distribution sample with unknown mean, test ${\sigma }^2={\sigma }_0^2$ against unrestricted positive variance. Put $q={\sum }_i(x_i-\bar{x}{)}^2/{\sigma }_0^2$ and assume $n\ge 2$. Profiling the mean and variance gives the likelihood ratio $\lambda (q)=(q/n{)}^{n/2}{e}^{(n-q)/2}$ for a nondegenerate sample. Its logarithmic derivative $\frac{1}{2}(n/q-1)$ changes sign at $n$, so small ratios select a lower and an upper tail.

Under the null hypothesis, Cochran's theorem gives $Q\sim {\chi }_{n-1}^2$. The exact size-$\alpha$ cutoffs $0<q_1<n<q_2$ therefore satisfyThe second equation makes the two boundary ratios equal. Rejection on $q\le q_1$ or $q\ge q_2$ is consequently the exact likelihood-ratio test, and its tails need not have equal probability. For a large sample, the Wilks theorem instead calibrates $W=-2\log \lambda$ by ${\chi }_1^2$. A Bartlett correction for a normal variance with unknown mean improves this asymptotic calibration; it is unnecessary when using the exact $Q$ distribution.