Invariant statistic
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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
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