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May, 1973 On Sufficiency and Invariance
D. Landers, L. Rogge
Ann. Statist. 1(3): 543-544 (May, 1973). DOI: 10.1214/aos/1176342420


Let $\mathscr{P}$ be a family of probability measures defined on a $\sigma$-field $\mathscr{A}$ on $X$ and $G$ be a group of transformations on $X$ such that $Pg^{-1} \in \mathscr{P}$ for all $P \in \mathscr{P}, g \in G$. Let $\mathscr{A}_I$ be the $\sigma$-field of $G$-invariant sets of $\mathscr{A}$ and $\mathscr{A}_{I^\ast}$ the $\sigma$-field of $\mathscr{P}$-almost $G$-invariant sets of $\mathscr{A}$. Let $\mathscr{A}_S$ be a sufficient $\sigma$-field for $\mathscr{P} \mid \mathscr{A}$. Hall, Wijsman and Ghosh proved that $\mathscr{A}_S \cap \mathscr{A}_I$ is sufficient for $\mathscr{P} \mid \mathscr{A}_I$ if $g\mathscr{A}_S = \mathscr{A}_S$ for each $g \in G$ and $\mathscr{A}_S \cap \mathscr{A}_I \sim \mathscr{A}_S \cap \mathscr{A}_{I^\ast}(\mathscr{P})$. They posed the question whether the first condition alone suffices to prove this result. An example shows that the answer is no. For dominated families we show that $\mathscr{A}_S \cap \mathscr{A}_{I^\ast}$ is always sufficient for $\mathscr{P} \mid \mathscr{A}_{I^\ast}$, a result which is not true any more for undominated families.


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D. Landers. L. Rogge. "On Sufficiency and Invariance." Ann. Statist. 1 (3) 543 - 544, May, 1973.


Published: May, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0258.62005
MathSciNet: MR378161
Digital Object Identifier: 10.1214/aos/1176342420

Primary: 65B05
Secondary: 62A05

Keywords: Invariance , Sufficient $\sigma$-fields

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 3 • May, 1973
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