Best Critical Region

Definition

Let $C$ : Subset of sample space

If

1. $P_{{\theta }^{\prime }}[\mathbf{X}\in C]=\alpha$
2. $\forall A\subset C,P_{{\theta }^{\prime }}[\mathbf{X}\in A]=\alpha ⟹P_{{\theta }^{\prime \prime }}[\mathbf{X}\in C]\ge P_{{\theta }^{\prime \prime }}[\mathbf{X}\in A]$

Then

• We say $C$ is a best critical region of size $\alpha$ for testing a simple statistical hypothesis $H_0:\theta ={\theta }^{\prime }$ against the alternative hypothesis $H_1:\theta ={\theta }^{\prime \prime }$
• We say the test of $C$ a best test

Related theorems

• Neyman-Pearson Theorem