Difference between revisions of "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 | + | <!-- |
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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|Invariant test]]; [[Invariance of a statistical procedure|Invariance of a statistical procedure]]). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Zacks, "The theory of statistical inference" , Wiley (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.P. Klimov, "Invariant inferences in statistics" , Moscow (1973) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Zacks, "The theory of statistical inference" , Wiley (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.P. Klimov, "Invariant inferences in statistics" , Moscow (1973) (In Russian)</TD></TR></table> | ||
Latest revision as of 22:13, 5 June 2020
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