Definition:Size (Inductive Statistics)
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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 $9.1.8$