Let $X_1,X_2,…,X_n$ be a random sample from any distribution with unknown parameter $\theta$ which takes values in a parameter space $\Theta$.

We ultimately want to test

\[H_0:\theta\in\Theta_0\\ H_1:\theta\in\Theta \backslash\Theta_0\]

where $\Theta_0$ is some subset of $\Theta$.

The Power Function

\[\begin{align} \gamma(\theta)&=P(\text{Reject $H_0$ when the parameter is $\theta$})\\ &=P(\text{Reject $H_0;\theta$}) \end{align}\]

$\theta$ is an argument that can be anywhere in the parameter space $\Theta$.

  • It could be a $\theta$ from $H_0$.
  • It could be a $\theta$ form $H_1$.

Note that

For $\alpha$:

\[\begin{align} \alpha&=\text{max}P(\text{Reject $H_0$ when true})\\ &=\underset{\theta\in\Theta_0}{\text{max}}P(\text{Reject $H_0;\theta$})\\ &=\underset{\theta\in\Theta_0}{\text{max}}\gamma(\theta)\\ &\big(\text{Other notation is $\underset{\theta\in H_0}{\text{max}}$}\big) \end{align}\]

For $\beta$:

\[\begin{align} \beta&=\text{max}P(\text{Fail to reject $H_0$ when false})\\ &=\underset{\theta\in\Theta\backslash\Theta_0}{\text{max}}P(\text{Fail to Reject $H_0;\theta$})\\ &=\underset{\theta\in\Theta\backslash\Theta_0}{\text{max}}\bigg[1 - P(\text{Reject $H_0;\theta$})\bigg]\\ &=\underset{\theta\in\Theta\backslash\Theta_0}{\text{max}}\bigg[1-\gamma(\theta)\bigg]\\ &\big(\text{Other notation is $\underset{\theta\in H_1}{\text{max}}$}\big) \end{align}\]

Why do we use power function?

  • They are great for comparing two hypothesis tests.