We are searching for a UMP. Let's assume $\theta_1 > \theta_0$.
When the test has the form: $H_o: \theta = \theta_o \hspace{10pt} H_a: \theta = \theta_1$ we can use Neyman-Pearson lemma to find the UMP.
When the test has the form: $H_o: \theta \leq \theta_o \hspace{10pt} H_a: \theta \gt \theta_0$ we can (often) use Karlin Rubin.
Now, I know we can also (often) use Karlin-Rubin for tests of the form: $H_o: \theta = \theta_o \hspace{10pt} H_a: \theta > \theta_0$
My question is this,
Under what conditions does Neyman-Pearson give the UMP for this test (Choosing any $\theta_1 > \theta_o$)
*Update
I found out that NP will give the same test, as long as the rejection region doesn't depend on $\theta_1$. I suppose an updated version of the question would be, can anybody give an example of a test, where the rejection region would depend explicitly on $\theta_1$?
