I know that the uniformly most powerful test is always unbiased, if it exists, but is the unbiased test always UMP?
2026-04-02 06:18:14.1775110694
Is the unbiased test always uniformly most powerful?
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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.