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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.

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  • $\begingroup$ 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 $n$. $\endgroup$ – StubbornAtom Dec 11 '19 at 5:50

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