Is the unbiased test always uniformly most powerful? [closed]
I know that the uniformly most powerful test is always unbiased, if it exists, but is the unbiased test always UMP?
No. Consider a randomized test that rejects the null with probability $\alpha$, independent of the data. Then the test has equal type 1 error and power and is unbiased level $\alpha$, but not necessarily UMP.