Abstract
We consider uniformly most powerful (UMP) as well as uniformly most powerful unbiased (UMPU) tests and their non-randomized versions for certain hypotheses concerning a binomial parameter. It will be shown that the power function of a UMP(U)-test based on sample size n can coincide on the entire parameter space with the power function of the corresponding test based on sample size n + 1. A complete characterization of this paradox will be derived. Apart some exceptional cases for two-sided tests and equivalence tests the paradox appears if and only if a test based on sample size n is non-randomized.
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