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:

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

Sources

• 2011: Morris H. DeGroot and Mark J. Schervish: Probability and Statistics (4th ed.): $9.1$: Problems of Testing Hypotheses: Definition $9.1.8$