Invariant statistic

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A statistic taking constant values on orbits generated by a group of one-to-one measurable transformations of the sample space. Thus, if is the sample space, is a group of one-to-one -measurable transformations of onto itself and is an invariant statistic, then for all and . Invariant statistics play an important role in the construction of invariant tests (cf. Invariant test; Invariance of a statistical procedure).


[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2] S. Zacks, "The theory of statistical inference" , Wiley (1971)
[3] G.P. Klimov, "Invariant inferences in statistics" , Moscow (1973) (In Russian)
How to Cite This Entry:
Invariant statistic. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098