Hypothesis testing have been extensively used on different discipline of science. And in this post, I will attempt on discussing the basic theory behind this, the Likelihood Ratio Test (LRT) defined below from Casella and Berger (2001), see reference 1. Definition . The likelihood ratio test statistic for testing H_0:\theta\in\Theta_0 versus H_1:\theta\in\Theta_0^c is \begin{equation} \label{eq:lrt} \lambda(\mathbf{x})=\frac{\displaystyle\sup_{\theta\in\Theta_0}L(\theta|\mathbf{x})}{\displaystyle\sup_{\theta\in\Theta}L(\theta|\mathbf{x})}. \end{equation}
A likelihood ratio test (LRT) is any test that has a rejection region of the form \{\mathbf{x}:\lambda(\mathbf{x})\leq c\}, where c is any number satisfying 0\leq c \leq 1. The numerator of equation (\ref{eq:lrt}) gives us the supremum probability of the parameter, \theta, over the restricted domain (null hypothesis, \Theta_0) of the parameter space \Theta, that maximizes the joint probability of the sample, $\math...