Invariant test

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A statistical test based on an invariant statistic. Let be a sampling space and suppose that the hypothesis : is tested against the alternative : , where the hypothesis is invariant under the group of one-to-one -measurable transformations of the space onto itself, that is,

where is the element of the group of one-to-one transformations of the probability measures : , defined for all and according to the formula , . Since is invariant under , in testing it is natural to use a test based on an invariant statistic with respect to this same group . 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 is invariant under the group , then there exists a maximin test in the class of invariant tests for testing . An invariant test is a special case of an invariant statistical procedure (see Invariance of a statistical procedure).


[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)
How to Cite This Entry:
Invariant test. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098