What is a most powerful test?
Suppose that you want to conduct a hypothesis test for the value of a parameter θ with a simple point hypothesis given as
H0: θ=θ0 vs H1: θ=θ1
Out of the many possible critical regions with size α, the test having the critical region with the greatest power among all of the others is called the most powerful test.
What is uniformly most powerful test?
Now suppose that you want to conduct a hypothesis test for the value of a parameter θ with a simple null hypothesis but a composite alternative hypothesis. A composite alternative hypothesis means that the parameter can take a range of values instead of a single point value.
H0: θ=θ0 vs H1: θ≠θ0
Out of the many possible critical regions with size α, the test having the critical region with the greatest power among all of the others is called the uniformly most powerful test.
Difference between most powerful and uniformly most powerful test:
The main difference is that a most powerful test has the greatest power among all tests of size α when the alternative hypothesis is a simple hypothesis (the parameter takes a single point value in a simple hypothesis).
In contrast, the uniformly most powerful test has the greatest power among all tests of size α when the alternative hypothesis is a composite hypothesis (the parameter can take a range of values under a composite hypothesis).
Uniformly Most Powerful tests and Likelihood Ratio:
It is known as a consequence of the Neyman Pearson lemma that the likelihood function gives us the most powerful tests when both hypotheses are simple. Sometimes, we can obtain uniformly most powerful tests for some composite hypotheses by treating a composite hypothesis as a collection of many simple hypotheses.
Formal Mathematical Definition of Uniformly Most Powerful Tests:
