Rejection Region

Definition

Let $\gamma$ be a test statistic, the subset of the outcome space for which the tests assumes the value 1 is referred to as the rejection region of the test. Formally, the rejection region is a subset of $\mathbb{R}$ defined as

$R=\{\gamma (y)\in \mathbb{R}|\varphi (y)=1\}.$

Remark. The random events $\varphi (y)=1$ and \gamma(y) \in R$$ are thus equivalent and associated with the same [[Test Statistic|probability under]] p_\theta$. In a concrete test scenario, it is hence usually the probability distribution of the test statistic that is of principal concern for assessing the test’s outcome behaviour.

Remark. In general, test decision rules considered are based on the test statistic exceeding a critical value $u\in R$. By means of the indicator function $1_{\gamma \ge u}$ , the tests considered here can thus be written as

\phi:&\mathbb{R}^n&\to&\{0,1\}\\ &y&\mapsto&\phi(y)=1_{\gamma\geq u}\;. \end{array}$$ ## References [[D. Ostwald - The general linear model 20_21|Ost20 pp. 117]]