I've been revising the past couple of days and have come across $2$ definitions of a UMP test.
Suppose we want to test $H_0: \theta=\theta_0$ vs $H_1: \theta>\theta_0$. Then the test is UMP if:
Definition $1)$ For each $\theta_1 \in \Theta_1$ (in this case $\theta_1>\theta_0$) we have that the optimal critical region $\boldsymbol{C}$ of the test $H_0: \theta=\theta_0$ vs $H_1: \theta=\theta_1$ is the same.
Definition $2)$ The power function $W(\theta)$ is maximised for each $\theta \in \Theta_1$.
Now I can see why $1) \implies 2)$. However, I'm a bit stuck on the proof of the other direction.
I know that $W(\theta_1)$ is the power of the test $H_0: \theta=\theta_0$ vs $H_1: \theta=\theta_1$ but can't really see why $W(\theta)$ being maximised for each $\theta \in \Theta_1$ implies that the critical regions $\boldsymbol{C}^{(\theta_1)}$ of the tests should all be the same. I was thinking whether it is not possible to come up with a counterexample or whether the proof itself uses NP - Lemma arguments.
Cheers!
