# Is a best critical region unique?

For testing a simple hypothesis $$H_0:\theta=\theta_0$$ against another simple hypothesis $$H_1: \theta=\theta_1$$, a best critical region or a most powerful test of size (aka, significance level) $$\alpha$$ is the region/test that maximizes the power function,i.e., minimizing the type II error, among all regions/tests with size $$\alpha$$.

Is a best critical region unique? I guess it is unique up to a set of probability zero, but I can't find any reference.

• A most powerful or a uniformly most powerful test is certainly not unique. For example, testing $H_0:\theta=\theta_0$ vs $H_1:\theta=\theta_1(>\theta_0)$ in a $U(0,\theta)$ distribution based on a sample of size $n$. – StubbornAtom Dec 11 '19 at 5:50