Power Function

Definition

We define power function of a critical region to be ${\gamma }_C(\theta )=P_{\theta }[(X_1,\dots ,X_n)\in C];\theta \in {\omega }_1$

Suppose $C_1,C_2$ : Critical regions, both of size $\alpha$

Then $C_1$ is better than $C_2$ if ${\gamma }_{C_1}(\theta )\ge {\gamma }_{C_2}(\theta ),\forall \theta \in {\omega }_1$