## 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.