The Annals of Statistics

On Sufficiency and Invariance

D. Landers and L. Rogge

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 1, Number 3 (1973), 543-544.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342420

Digital Object Identifier
doi:10.1214/aos/1176342420

Mathematical Reviews number (MathSciNet)
MR378161

Zentralblatt MATH identifier
0258.62005

JSTOR
links.jstor.org

Subjects
Primary: 65B05: Extrapolation to the limit, deferred corrections
Secondary: 62A05

Keywords
Sufficient $\sigma$-fields invariance

Citation

Landers, D.; Rogge, L. On Sufficiency and Invariance. Ann. Statist. 1 (1973), no. 3, 543--544. doi:10.1214/aos/1176342420. https://projecteuclid.org/euclid.aos/1176342420


Export citation