Q:

Likelihood-ratio test

I was studying on this subject and I got some questions. Lets take the
  test $$H_0:\theta\in\Theta_0 \space vs\space H_1:\theta\in\Theta_0^c$$
  where $\Theta$ is the parametric space and
  $$\lambda(x)=\frac{\sup_{\Theta_0}L(\theta|x)}{\sup_{\Theta^c}L(\theta|x)}$$
  and the critical region is:Reject $H_0$ if {$\lambda(x)\leq c$} where
  $0\leq c\leq 1$

No matter what the hypotheses, the critical region is always that way?
I choose $c$ such that $P_\theta(\lambda(x)\leq c)=\alpha$? Where $\alpha$ is the size of test.

A:

The Likelihood Ratio Test is an intuitive construction. It measures the cost of restricting the space of maximization in terms of maximized likelihoods. Note that the ratio will always be below  $1$ as the unrestricted maximization will always result in greater likelihood. Hence an intuitive rejection rule would be to reject the null hypothesis if the ratio becomes too small. Here is the idea

One question would be "how small is small". Well this is why we need the distribution of the statistic. It can actually be shown that if a sufficient statistic exists, the LRT will boil down to a test based on it.  Even then, however, it might not be easy to obtain the exact distribution. In those cases, the asymptotic $\chi^2$ distribution of $-2 \log \lambda(x)$ comes in handy. The asymptotic approximation is often referred to as Wilk's theorem by the way.
This is the LRT in a few words. Many useful references can be found via the advanced act of google search :)

