Further Discussion
Finally, we note that in general, UMP tests do not exist for vector parameters or for two-sided tests (a test in which one hypothesis lies on both sides of the alternative). Why is it so?
The reason is that in these situations, the most powerful test of a given size for one possible value of the parameter (e.g. for where ) is different from the most powerful test of the same size for a different value of the parameter (e.g. for where ). As a result, no test is uniformly most powerful.
Read more about this topic: Uniformly Most Powerful Test
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