Best Critical Region

Definition

Let $C$ be a subset of sample space. $C$ is a best critical region of size $\alpha$ for testing the simple hypothesis test $H_0:\theta ={\theta }^{\prime },H_1:\theta ={\theta }^{\prime \prime }$ if

• $P_{{\theta }^{\prime }}[X\in C]=\alpha$ (level alpha condition)
• $\forall A,P_{{\theta }^{\prime }}(X\in \alpha )\Rightarrow P_{{\theta }^{\prime \prime }}(X\in C)\ge P_{{\theta }^{\prime \prime }}(X\in A)$ (most powerful)