\[\Lambda \left( x \right) = \frac{ sup \{ L(\theta | x) : \theta \in \Theta_0 \}}{sup \{ L(\theta | x) : \theta \in \Theta \}}\]
The denominator has \(\Theta\) instead \(\Theta_0^C\) because value of \(\Lambda\) is pretty similar and its properties are much simpler to derive.
Thus, the rejection region is
\[R=\{x: \Lambda(x)<c\}\]
If the null hypothesis holds
\[D = - 2 \log{\Lambda} \xrightarrow{d} \chi^2_p\]
where \(D\) is called the deviance and \(p\) is the number of constrained parameters.
\[\Lambda \left( x \right) = \frac{L(\widehat{\theta}_0 | x)}{L(\widehat{\theta}_1 | x)}\]
The test \(\Lambda(x)<c\) with \(P_{\theta_0}(x \in R) =\alpha\) is the UMP test.