Lecture 11. E) Power Function

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Example 1 >>

Power Function

The power function of a test with critical region [math]C[/math] is the function [math]\beta:\Theta\rightarrow\left[0,1\right][/math] given by

[math]\beta\left(\theta\right)=P_{\theta}\left[\left(X_{1}..X_{n}\right)'\in C\right]=P_{\theta}\left(\text{reject }H_{0}\right),\theta\in\Theta.[/math]

So, the power function returns a probability. It is super convenient because it summarizes type 1 and type 2 errors in a single function.

To see this, note that

[math]\begin{aligned} P_{\theta}\left(\text{type 1 error}\right) & =\beta\left(\theta\right),\,\theta\in\Theta_{0}\\ P_{\theta}\left(\text{type 2 error}\right) & =1-\beta\left(\theta\right),\,\theta\in\Theta_{1}\end{aligned}[/math]

From here on, we will work with examples.

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