In posing a statistical problem
one specifies a set X, a σ-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.
