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January, 1973 Nonlinear Operators and Sufficiency
J. Pfanzagl, J. Weber
Ann. Statist. 1(1): 183-187 (January, 1973). DOI: 10.1214/aos/1193342398


Let $(X, \mathscr{A})$ be a measurable space and $\mathscr{P} \mid \mathscr{A}$ a family of probability measures. Given a sufficient sub-$\sigma$-field, we define the sufficiency operator by assigning to each integrable function a conditional expectation which is independent of $P \in \mathscr{P}$. The operator thus defined is $\mathscr{P}$-a.e. linear, monotone, idempotent, and expectation invariant for every $P \in \mathscr{P}$. Not all of these properties are necessary to make such an operator useful in statistical theory: For the most important applications, linearity may be replaced by homogeneity and translation invariance; idempotency may be relinquished. It was therefore suggested (Pfanzagl (1967), page 416) to study homogeneous, translation invariant, monotone, and expectation invariant operators as a possibly useful generalization of sufficiency-operators. The purpose of this paper is to show that there exists to any such operator a sufficient sub-$\sigma$-field whose sufficiency operator effects at least the same reduction. (Hence there is no real gain in introducing the idea of "reduction by homogeneous, translation invariant, monotone, and expectation invariant operators" into statistical theory.) This result generalizes Proposition 9 of LeCam (1964), page 1435, where, roughly speaking, a similar result was obtained for monotone and expectation invariant operators which are linear (rather than homogeneous and translation invariant). An ergodic lemma for homogeneous and translation invariant (but not necessarily linear) operators, needed here as a tool for the proof of the main theorem, may be of independent interest.


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J. Pfanzagl. J. Weber. "Nonlinear Operators and Sufficiency." Ann. Statist. 1 (1) 183 - 187, January, 1973.


Published: January, 1973
First available in Project Euclid: 25 October 2007

zbMATH: 0253.62006
MathSciNet: MR336859
Digital Object Identifier: 10.1214/aos/1193342398

Keywords: 28 A65 , 62 B05 , ergodic theorem , nonlinear operators , sufficiency

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 1 • January, 1973
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