Suppose we sample from a population with a distribution that is completely specified except for the value of a single parameter . If we wish to test (simple) versus (composite) there is no general theorem like Theorem 3.1 that can be applied. But it can be applied to find a MP test for versus for any single value . In many situations the form of the rejection region for the MP test does not depend on the particular choice of . When a test obtained by Theorem 1.3 actually maximizes the power for every value of , it is said to be uniformly most powerful, (UMP) for against .
We may state the definition as follows:
The critical region C is a uniformly most powerful critical (UMPCR) of size for testing the simple hypothesis against a composite alternative if the set C is a best critical region of size for testing against each simple hypothesis in . A test defined by this critical region C is called a uniformly most powerful test, with significance level , for testing the simple against the composite .
Uniformly most powerful tests don't always exist, but when they do, the Neyman Pearson Theorem provides a technique for finding them.
Let be a random sample from a distribution where the variance is unknown. Find a UMP test for () against .
likelihood of the sample is
So defined above is a BCR of size for testing against . We note that, for each number , the above argument holds. So is a UMP critical region of size for testing against . To be specific, suppose now that , , , show that .