I am struggling a little to understand the difference between these two classes of tests.
Suppose we were testing a simple null hypothesis and a composite two sided alternative hypothesis. I am under the impression that it is impossible to construct a UMP test for these hypotheses, however it is possible to construct a UMPU test for these hypotheses.
The way I understand it is that the critical region given by Neyman-Pearson changes either side of the tail hence won't be most powerful for any particular value of theta.
Here is where my confusion lies: taking the union of critical regions for either tail isn't UMP. I think it is because if we have to restrict each critical region to alpha/2 then, say we know it is in one tail, we are wasting part of the critical region on a tail X1,...,Xn is less likely to fall in and so there is a more optimal test putting more of the critical region on the other tail. Is this correct?
Now if this correct, I cannot fathom how restricting the critical regions to unbiased tests (i.e. power under H1>size) fixes this problem.