Multiparametric Likelihood Ratio Test

Definition

Let $X_1,X_2,\dots ,X_n$ be Random Sample from PDF $f(x|\mathbf{\theta })$, where $\mathbf{\theta }\in \Omega \subset {\mathbb{R}}^p$, $p$ be the number of parameters, $q$ be the number of constraints (restrictions) $g_1(\mathbf{\theta })=a_1,\dots ,g_q(\mathbf{\theta })=a_q$, where $g_i$ is differentiable function $H_0:\mathbf{\theta }\in \omega ,H_1:\mathbf{\theta }\in \Omega \setminus \omega$, where the dimension of $\omega$ is $p-q$.

Now, consider a test statistic to test the hypothesis $\Lambda :=\frac{\underset{\theta \in \omega }{\max }L(\theta )}{\underset{\theta \in \Omega }{\max }L(\theta )}=\frac{L(\hat{\omega })}{L(\hat{\Omega })}$ If $H_0$ is true, then $\Lambda$ will be close to $1$, while if $H_0$ is not true, then the $\Lambda$ will be close to $0$ Therefore, LRT rejects $H_0$ when $\Lambda \le c$, where the $c$ is determined by the level alpha condition. $\alpha =P_{\omega }(\Lambda \le c)$

By the Wilks’ Theorem, the likelihood ratio test Statistic for the hypothesis $H_0:\mathbf{\theta }\in \omega ,H_1:\mathbf{\theta }\in \Omega \setminus \omega$ is given by ${\chi }_L^2:=-2\ln \Lambda =-2\ln \frac{L(\hat{\omega })}{L(\hat{\Omega })}\stackrel{D}{\to }{\chi }^2(q)$