On the Order Structure of the Set of Sufficient Subfields

Abstract

In [5], the concept of statistical sufficiency is studied within a general probability setting. The study is continued here. The notation and definitions of [5] are used. Here we give an example of sufficient statistics t ₁ and t ₂ such that the pair (t ₁ , t ₂ ) is not sufficient. The example also has the property that, in a sense to be made precise, no smallest sufficient statistic containing t ₁ and t ₂ exists. In Example 4 of [5], sufficient subfields A₁and A₂are exhibited such that A₁ ۷ A₂, the smallest subfield containing A₁and A₂, is not sufficient. Such an example is given here with the even stronger property that no smallest sufficient subfield containing A₁ and A₂ exists.

Received Octomber 30, 1961.