Similarity region

similar region

A generally used abbreviation of the term "critical region similar to a sample space" as used in mathematical statistics for a critical region with non-randomized similarity of a statistical test.

Let $X$ be a random variable taking values in a sample space $(\mathfrak{X},\mathfrak{B},{\mathsf{P}}_{\theta })$, $\theta \in \mathrm{\Theta }$, and consider testing the compound hypothesis $H_0$: $\theta \in {\mathrm{\Theta }}_0\subset \mathrm{\Theta }$ against the alternative $H_1$: $\theta \in {\mathrm{\Theta }}_1=\mathrm{\Theta }\setminus {\mathrm{\Theta }}_0$. Suppose that in order to test $H_0$ against $H_1$, a non-randomized similar test of level $\alpha$( $0<\alpha <1$) has been constructed, with critical function $\varphi (x)$, $x\in \mathfrak{X}$. As this test is non-randomized,

$$\begin{array}{}\text{(1)}& \varphi (x)=\{\begin{array}{ll}1,& x\in K\subset \mathfrak{X},\\ 0,& x\notin K,\end{array}\end{array}$$

where $K$ is a certain set in $\mathfrak{X}$, called the critical set for the test (according to this test, the hypothesis $H_0$ is rejected in favour of $H_1$ if the event $\{X\in K\}$ is observed in an experiment). Also, the constructed test is a similar test, which means that

$$\begin{array}{}\text{(2)}& \underset{\mathfrak{X}}{\int }\varphi (x)d{\mathsf{P}}_{\theta }=\alpha \text{ for }\text{ all }\theta \in {\mathrm{\Theta }}_0.\end{array}$$

It follows from (1) and (2) that the critical region $K$ of a non-randomized similar test has the property:

$${\mathsf{P}}_{\theta }\{X\in K\}=\alpha \text{ for }\text{ all }\theta \in {\mathrm{\Theta }}_0.$$

Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called $K$ a "region similar to the sample space" $\mathfrak{X}$, in the sense that the two probabilities ${\mathsf{P}}_{\theta }\{X\in K\}$ and ${\mathsf{P}}_{\theta }\{X\in \mathfrak{X}\}$ are independent of $\theta \in {\mathrm{\Theta }}_0$.

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