Definition:Size (Inductive Statistics)

Definition
Let $\theta$ be a population parameter of some population.

Let $\Omega$ be the parameter space of $\theta$.

Let $\Omega_0$ and $\Omega_1$ be disjoint subsets of $\Omega$ such that $\Omega_0 \cup \Omega_1 = \Omega$.

Let $\delta$ be a test procedure of the hypotheses:


 * $H_0: \theta \in \Omega_0$
 * $H_1: \theta \in \Omega_1$

Let $\pi$ be the power function of $\delta$.

The size of $\delta$, usually denoted $\alpha$, is defined as:


 * $\ds \alpha = \sup_{\theta \in \Omega_0} \map \pi \theta$