Under the assumption that $H_0$ is true, where $\gamma(\vec X)$ is the Generalized Likelihood Ratio from the [[Generalized Likelihood Ratio Test]], $-2 \ ln \ \gamma(\vec X) \overset{d}{\to} \chi^2(1)$ In other words, $-2 \ln \gamma(\vec X)$ [[converges in distribution]] to the [[chi-squared distribution]] with $1$ [[degrees of freedom]]. However, Wilks' Theorem does not hold with the parameter is involved in the [[support]] of the [[probability density function|pdf]]. Suppose that $X_1, X_2, \dots, X_n$ is a random sample from a distribution with pdf $f(x; \theta)$ where $\theta$ is not involved in the support of $f$ and $n$ is large. An approximate GLRT of size $\alpha$ for testing $\begin{align} H_0: \theta = \theta_0 && H_1: \theta \ne \theta_0 \end{align}$ is to reject $H_0$ if $-2 \ln \ \gamma(\vec X) > \chi^2_{\alpha, 1}$