Definition:Critical Region

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 $\delta$.

The critical region is also referred to as the rejection region