Nonlinear Operators and Sufficiency

Abstract

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.