A more general property than domination for sets of probability measures

Abstract

In posing a statistical problem one specifies a set X, a field S of subsets of X, and a collection M of probability measures on (X, S). It is often convenient to impose some condition on M in order to avoid measure theoretic difficulties and the condition most often used is domination, i.e., the existence of a probability measure with respect to which each of the measures in M is absolutely continuous. In this paper we introduce a more general condition, which we call compactness, implying the existence of a best sufficient subfield and of certain estimates. It is also possible to characterize, under this condition, those functions on M admitting unbiased estimates of certain types.