## The Annals of Mathematical Statistics

### A Note on Sufficiency and Invariance

Robert H. Berk

#### Abstract

Under certain conditions, it is shown that the invariant and almost-invariant $\sigma$-fields are equivalent if and only if the invariant $\sigma$-field is independent of an appropriate sufficient $\sigma$-field. This result helps unify work of Hall, Wijsman and Ghosh and of Pfanzagl, who dealt with the forward implication and work of Berk and Bickel, who treated the reverse implication. The conditions required are that the sufficient and invariant $\sigma$-fields be essentially disjoint and together generate the $\sigma$-field of the original data.

#### Article information

Source
Ann. Math. Statist., Volume 43, Number 2 (1972), 647-650.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177692645

Digital Object Identifier
doi:10.1214/aoms/1177692645

Mathematical Reviews number (MathSciNet)
MR298810

Zentralblatt MATH identifier
0254.62002

JSTOR