Power of a Test

Definition

Let $\theta \in {\omega }_1$¹

Then

• $1-P_{\theta }[\text{Type II Error}]=P_{\theta }[(X_1,\dots ,X_n)\in C]$
• We say $P_{\theta }[(X_1,\dots ,X_n)\in C]$ the power of the test at $\theta$

Remark

Note that $P_{\theta }[(X_1,\dots ,X_n)\in C]$ should be read as the probability that $(X_1,\dots ,X_n)\in C$ when $\theta$ is the true parameter.

Over all critical regions of size $\alpha$, we want to consider those which have lower probabilities of Type II error.

We also can look at the complement of a Type II error, i.e., rejecting $H_0$ when $H_1$ is true, which is a correct decision. Since we want the probability of this to be as large as possible, therefore we need to maximize the power of the test at $\theta$.

Minimizing the probability of Type II error is equivalent to maximizing power.

Related

• Power Function

Footnotes

1. See Hypothesis ↩