Power function of a statistical test

For a test with rejection region $R$, its power function isIts supremum over the null parameter space is the size of the test.

Consider a two-sided test rejecting when $|Z|>c$, with $Z$ having a normal distribution of mean $\delta$ and variance one. Here $\Phi$ is the cumulative distribution function of the standard normal distribution. Under a nonzero alternative $\delta$, the type II error probability is the displayed function. It tends to the nonrejection probability under the null as $\delta \to 0$, and to zero as $\delta \to +\infty$. A particular observed $Z=z$ has positive density under all these alternatives. Its null-calibrated p-value therefore does not select one value of $\beta$. The power function describes repeated-sampling performance at specified alternatives.