Invariant test

A statistical test based on an invariant statistic. Let $ ( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta , \theta \in \Theta ) $ be a sampling space and suppose that the hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subset \Theta $ is tested against the alternative $ H _ {1} $: $ \theta \in \Theta _ {1} = \Theta \setminus \Theta _ {0} $, where the hypothesis $ H _ {0} $ is invariant under the group $ G = \{ g \} $ of one-to-one $ \mathfrak B $- measurable transformations of the space $ \mathfrak X $ onto itself, that is,

$$ \overline{g}\; \Theta _ {0} = \Theta _ {0} \ \textrm{ and } \ \ \overline{g}\; \Theta _ {1} = \Theta _ {1} \ \textrm{ for } \textrm{ any } \ g \in G , $$

where $ \overline{g}\; $ is the element of the group $ \overline{G}\; = \{ \overline{g}\; \} $ of one-to-one transformations of the probability measures $ {\mathsf P} _ \theta $: $ {\mathsf P} _ \theta \rightarrow {\mathsf P} _ {\overline{g}\; \theta } $, defined for all $ \theta \in \Theta $ and $ g \in G $ according to the formula $ {\mathsf P} _ {\overline{g}\; \theta } ( B ) = {\mathsf P} _ \theta ( g ^ {-} 1 B ) $, $ B \in \mathfrak B $. Since $ H _ {0} $ is invariant under $ G $, in testing $ H _ {0} $ it is natural to use a test based on an invariant statistic with respect to this same group $ G $. Such a test is called an invariant test, and the class of all invariant tests is the same as the class of tests based on a maximal invariant. In the theory of invariant tests, the Hunt–Stein theorem plays an important role: If the hypothesis $ H _ {0} $ is invariant under the group $ G $, then there exists a maximin test in the class of invariant tests for testing $ H _ {0} $. An invariant test is a special case of an invariant statistical procedure (see Invariance of a statistical procedure).

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1988)
[2]	L. Schmetterer, "Introduction to mathematical statistics" , Springer (1974) (Translated from German)
[3]	W.J. Hall, R.A. Wijsman, J.K. Chosh, "The relationships between sufficiency and invariance with applications in sequential analysis" Ann. Math. Stat. , 36 (1965) pp. 575
[4]	G.P. Klimov, "Invariant inferences in statistics" , Moscow (1973) (In Russian)
[5]	S. Zacks, "The theory of statistical inference" , Wiley (1971)