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 be a random variable taking values in a sample space , , and consider testing the compound hypothesis : against the alternative : . Suppose that in order to test against , a non-randomized similar test of level () has been constructed, with critical function , . As this test is non-randomized,

(1)

where is a certain set in , called the critical set for the test (according to this test, the hypothesis is rejected in favour of if the event is observed in an experiment). Also, the constructed test is a similar test, which means that

(2)

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

Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called a "region similar to the sample space" , in the sense that the two probabilities and are independent of .

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