Abstract
Given two simple hypotheses concerning the value of the parameter on which the distribution function of a random variable depends, Neyman-Pearson's lemma gives the form of the most powerful test of dimension alfa(existence) and, under certain conditions, the lemma also assures the uniqueness of the most powerful test. This note gives a necessary and sufficient condition for the non uniqueness of the most powerful test, that is for the existence of tests different from Neyman-Pearson's, but characterized by the same dimension and power.
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