Wilks' theorem

Remarks

The number $q$ is the difference between the dimension of $\Theta$ and the dimension of $\Theta_0$. Proof idea: expand the log-likelihood to second order around the unrestricted and the restricted maximum likelihood estimates. The difference becomes a quadratic form in an asymptotically normal vector, which has a limiting chi-squared law with $q$ degrees of freedom. Example: for an i.i.d. $\mathcal{N}(\mu, \sigma^2)$ sample and $H_0 \colon \mu = 0$, here $k = 2$ and $q = 1$, so one rejects when $-2 \log \lambda_n$ exceeds the $1 - \alpha$ quantile of $\chi^2_1$. The hypothesis that the true parameter is an interior point matters. For a variance component tested at the boundary value $0$, the limit law is a mixture of $\chi^2_0$ and $\chi^2_1$ rather than $\chi^2_1$.

Going deeper

For a student

This is why likelihood-ratio tests are used everywhere: one computes $-2 \log \lambda_n$ and compares it with a chi-squared quantile. The score test and the Wald test have the same limiting chi-squared law under the null hypothesis, and the three tests are asymptotically equivalent. The theorem needs regularity: smooth densities, identifiability, and a true parameter in the interior of the parameter set. It is an asymptotic statement, so for small samples the chi-squared approximation can be poor.