Power function of a test

The power function of a test of $H_0:\theta \in {\mathrm{\Theta }}_0$ against $H_a:\theta \in {\mathrm{\Theta }}_1$

The power function is the probability of rejecting the $H_0$ for a given value of $\theta$, $\pi (\theta )=P(\text{reject }{H}_0|\theta )$

• at a single point, we can find the probability that our test will reject the null, $\alpha$ when $H_0$ is true, and $1-\beta$ when $H_a$ is true

Select a critical region that has a small $\alpha$ and largest power $(1-\beta )$ for the specific test being done

If a test satisfies $P(\text{reject }{H}_0|H_0\text{ is true})\le \alpha$ then the test is called an $\alpha$ level significant test

An $\alpha$ level significant test with the smallest $\beta$ is called the uniformly most powerful test (UMPT)

When testing a simple null versus a simple alternative the N-P lemma gives the most powerful test