A note on hypothesis testing for discrete cases

Abstract

Over the years, students have been taught to rely solely on the Neyman-Pearson Lemma in the hypothesis testing problem of a simple hypothesis against a simple alternative hypothesis to obtain a most powerful test, or equivalently, a best critical region of size β. If a particular β level is not attainable by the lemma, randomization can be employed. However, randomized tests are impractical in reality. In this note, illustrations are given to show that most powerful tests may still be found in discrete situations where the Neyman-Pearson Lemma does not apply and randomized tests are not adopted.