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, $\mathbf{x}$. While the denominator of the LRT gives us the supremum probability of the parameter, $\theta$, over the unrestricted domain, $\Theta$, that maximizes the joint probability of the sample, $\mathbf{x}$. Therefore, if the value of $\lambda(\mathbf{x})$ is small such that $\lambda(\mathbf{x})\leq c$, for some $c\in [0, 1]$, then the true value of the parameter that is plausible in explaining the sample is likely to be in the alternative hypothesis, $\Theta_0^c$.Definition. Thelikelihood ratio test statisticfor 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} Alikelihood 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 $0\leq c \leq 1$.

**Example 1**. Let $X_1,X_2,\cdots,X_n\overset{r.s.}{\sim}f(x|\theta)=\frac{1}{\theta}\exp\left[-\frac{x}{\theta}\right],x>0,\theta>0$. From this sample, consider testing $H_0:\theta = \theta_0$ vs $H_1:\theta<\theta_0$.