Vol. 15, No. 2, 1965

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A more general property than domination for sets of probability measures

Tom Stephen Pitcher

Vol. 15 (1965), No. 2, 597–611

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.

Mathematical Subject Classification
Primary: 46.90
Received: 17 July 1963
Revised: 31 March 1964
Published: 1 June 1965
Tom Stephen Pitcher