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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 $ ( \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)
How to Cite This Entry:
Invariant statistic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invariant_statistic&oldid=47418
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article