Invariant statistic

A statistic taking constant values on orbits generated by a group of one-to-one measurable transformations of the sample space. Thus, if $ ( \mathfrak X , \mathfrak B ) $ is the sample space, $ G = \{ g \} $ is a group of one-to-one $ \mathfrak B $- measurable transformations of $ \mathfrak X $ onto itself and $ t ( x) $ is an invariant statistic, then $ t ( gx ) = t ( x) $ for all $ x \in \mathfrak X $ and $ g \in G $. Invariant statistics play an important role in the construction of invariant tests (cf. Invariant test; Invariance of a statistical procedure).

References

[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)