Neyman-Pearson lemma

In statistics, the Neyman-Pearson lemma states that when performing a hypothesis test between two point hypotheses H0: θ=θ0 and H1: θ=θ1, then the likelihood-ratio test which rejects H0 in favour of H1 when


 * $$\Lambda(x)=\frac{ L( \theta _{0} \mid x)}{ L (\theta _{1} \mid x)} \leq \eta \mbox{ where } P(\Lambda(X)\leq \eta|H_0)=\alpha$$

is the most powerful test of size &alpha; for a threshold η. If the test is most powerful for all $$\theta_1 \in \Theta_1$$, it is said to be uniformly most powerful (UMP).

In practice, the likelihood ratio itself is not actually used in the test. Instead one computes the ratio to see how the key statistic in it is related to the size of the ratio (i.e. whether a large statistic corresponds to a small ratio or to a large one).