ABSTRACT
This chapter studies the relationship between sufficiency and ancillarity in greater detail, paying particular attention to minimal sufficiency and maximal ancillarity. Theorem One in Basu states that any ancillary statistic is independent of a sufficient statistic, in the sampling process, provided a further condition, which links the parameter and the sufficient statistic is satisfied, namely, the sufficient statistic must be boundedly complete. The chapter generalizes two notions, measurable separability and strong identifiability so as to provide similar results in reduced experiments. These two properties are of some independent interest. The chapter presents a general theory of these properties and of their relationship with both conditional independence and the projection of σ-fields. The concept of a measurably separated Bayesian experiment and then analyse the role of measurable separability in Basu Second Theorem both in a purely Bayesian form and in sampling theory. The concept of strong identification may be interpreted in terms of the geometry of Banach spaces.
