Neyman Factorization and Minimality of Pairwise Sufficient Subfields

Abstract

Assume that every probability measure $P$ in $\mathscr{P}$ of a statistical structure $(X, \mathscr{A}, \mathscr{P})$ has a density $p(x, P)$ w.r.t. a (not necessarily $\sigma$-finite) measure $m$. Let $\mathscr{B}$ be any subfield and suppose that the densities are factored as $p(x, P) = g(x, P)h(x)$ where $g$ is $\mathscr{B}$-measurable. Then $\mathscr{B}$ is pairwise sufficient and contains supports of $P$'s. Assume further that $m$ is locally localizable and $\mathscr{B}$ is pairwise sufficient and contains supports of $P$'s. Then the densities are factored as above. Two partial orders are introduced for pairwise sufficient subfields. Assuming that every $P$ has a support, a subfield is constructed which is the smallest with supports under the first partial order, and is the smallest under the second. This is used to give a simple proof of existence of the minimal sufficient subfield for the coherent case. In the (uncountable) discrete case it is proved that under the first partial order there are infinitely many minimal pairwise sufficient subfields and hence there is none that is smallest.