For composite hypothesis tests

\[\Lambda \left( x \right) = \frac{ sup \{ L(\theta | x) : \theta \in \Theta_0 \}}{sup \{ L(\theta | x) : \theta \in \Theta \}}\]

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

\[R=\{x: \Lambda(x)<c\}\]

Wilks theorem

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.

For simple hypothesis test

\[\Lambda \left( x \right) = \frac{L(\widehat{\theta}_0 | x)}{L(\widehat{\theta}_1 | x)}\]

Neyman-Pearson lemma

The test \(\Lambda(x)<c\) with \(P_{\theta_0}(x \in R) =\alpha\) is the UMP test.