Definition:Critical Region
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Definition
Let $\theta$ be a population parameter of some population $P$.
Let $\Omega$ be the parameter space of $\theta$.
Let $\mathbf X$ be a random sample from $P$.
Let $T = \map f {\mathbf X}$ be a sample statistic.
Let $\delta$ be a hypothesis test of the form:
- reject $H_0$ if $T \in C$
for some null hypothesis $H_0$ and some $C \subset \Omega$.
We refer to $C$ as the critical region of $\delta$.
This critical region is specifically chosen so as to make it so that the $p$-value is greater than the required significance level for the given test $\delta$.
Also known as
The critical region is also referred to as the rejection region
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hypothesis testing
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): critical region
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hypothesis testing
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): critical region
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hypothesis testing
- 2011: Morris H. DeGroot and Mark J. Schervish: Probability and Statistics (4th ed.): $9.1$: Problems of Testing Hypotheses: Definition $9.1.4$