Characterizing Non-nesting for the Neyman-Pearson Family of Tests

Abstract

For testing a simple null hypothesis against a simple alternative using Neyman-Pearson theory, examples of most powerful non-randomized critical regions are constructed, which are overlapping for varying sizes. A likelihood ratio based criterion, characterizing such critical regions, is also provided. A simple method, in addition, is suggested to construct the class of distributions providing overlapping critical regions for unequal sizes. These examples, in fact, counterexamples are important in explaining the fact that power of an optimum test may not increase with an increase in size.

Keywords

• Nested critical region; Most Powerful test