Likelihood ratio

The likelihood ratio for $$\theta = \theta_1$$ relative to $$\theta = \theta_0$$ is given by

$$LR = \frac{L(\theta_1)}{L(\theta_0)}$$

It is usually easier to work with log-likelihood, which allow the mutiplicitve relationship to be worked with additively, which suggests a log-likelihood ratio like so;

$$log( LR ) = log( \frac{L(\theta_1)}{L(\theta_0)}) = \ell(\theta_1) - \ell(\theta_0)$$

If LR is greater than 1, there is evidence for 1 If LR is less than 1, there is evidence for 0

The likelihood ratio test
This provides a test

$$H_0:\theta = \theta_0$$ against $$H_1:\theta = \theta_1$$

which has a rejection region

and some critical value

Neyman–Pearson Lemma
this lemma states that when performing a hypothesis test between two point hypothesese, then the likelihood-ratio test s the most powerful test of size \alpha for a threshhold n

The importance of the Neyman–Pearson Lemma derives from the fact that it provides a general method for devising a most powerful test of two simple hypotheses.