Wilks' Theorem

Definition

Single Parameter

Under R0 ~ R5 regularity conditions, let $H_0:\theta ={\theta }_0$, $-2\ln \Lambda \stackrel{D}{\to }{\chi }^2(1)$ where $\Lambda =\frac{L({\theta }_0)}{L(\hat{\theta })}$ is the likelihood ratio, and the $\hat{\theta }$ is the MLE of $\theta$

Multiparameter

Under R0 ~ R5 regularity conditions, let $H_0:\mathbf{\theta }\in \omega$, $-2\ln \Lambda \stackrel{D}{\to }{\chi }^2(q)$ where $\Lambda =\frac{L(\hat{\omega })}{L(\hat{\Omega })}$ is the likelihood ratio, and the $\hat{\Omega }$ is the MLE of $\mathbf{\theta }$ without constraints.