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 \Theta $, and consider testing the compound hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subset \Theta $ against the alternative $ H _ {1} $: $ \theta \in \Theta _ {1} = \Theta \setminus \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 $ \phi ( x) $, $ x \in \mathfrak X $. As this test is non-randomized,

$$ \tag{1 } \phi ( x) = \ \left \{ \begin{array}{ll} 1, &{ x \in K \subset \mathfrak X, } \\ 0, &{ x \notin K, } \\ \end{array} \right . $$

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

$$ \tag{2 } \int\limits _ { \mathfrak X } \phi ( x) d {\mathsf P} _ \theta = \alpha \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . $$

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

$$ {\mathsf P} _ \theta \{ X \in K \} = \alpha \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . $$

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 \Theta _ {0} $.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2]	B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[3]	J. Neyman, E.S. Pearson, "On the problem of the most efficient tests of statistical hypotheses" Philos. Trans. Roy. Soc. London Ser. A , 231 (1933) pp. 289–337
[4]	E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation I" Sankhyā , 10 (1950) pp. 305–340
[5]	E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation II" Sankhyā , 15 (1955) pp. 219–236