ABSTRACT
In Chapter 8, the Neyman-Pearson theory of uniformly most powerful (UMP) level https://www.w3.org/1998/Math/MathML"> α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq8361.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> tests was developed. We recall that even in a one-parameter situation, sometimes a UMP level https://www.w3.org/1998/Math/MathML"> α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq8362.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> test does not exist for deciding between a simple null and a two-sided alternative hypothesis. In Section 8.5.1, we cited a testing problem for the mean of a normal distribution (with known variance) when the alternative hypothesis was two-sided! In many situations where a UMP test cannot be found, likelihood ratio tests often save the day by providing an indispensable statistical tool. A general approach for testing composite null and alternative hypotheses was developed by Neyman and Pearson (1928a, b, 1933a,b).
