Power of a test

Remarks

For $\theta \in \Theta_1$, the power is $1 - \beta(\theta)$, where $\beta(\theta)$ is the probability of a type II error. For $\theta \in \Theta_0$, the value $\pi(\theta)$ is the probability of a type I error, and $\sup_{\theta \in \Theta_0} \pi(\theta)$ is the size of the test. For a randomized test $\varphi$ with values in $[0, 1]$, the power function is $\pi(\theta) = E_\theta[\varphi(X)]$. Example: let $X \sim \mathcal{N}(\theta, 1)$, $H_0 \colon \theta = 0$, $H_1 \colon \theta > 0$, and $R = \set{x > 1.645}$. Then $\pi(\theta) = 1 - \Phi(1.645 - \theta)$, where $\Phi$ is the standard normal distribution function; so $\pi(0) \approx 0.05$ and $\pi(2) \approx 0.64$. Non-example: $\pi(0) \approx 0.05$ is not a power in this example, since $0 \notin \Theta_1$; it is the size.