Generalizations of Ancillarity, Completeness and Sufficiency in an Inference Function Space

In this paper we introduce E-ancillarity and complete E-sufficiency, natural extensions of the definitions of ancillarity and complete sufficiency to a space of estimating or inference functions. These are functions of both the data and the parameter. We begin either with a space of all such functions or with a subset defined to exploit special features of a model; for example, we allow restrictions to inference functions that are linear in the observations or linear in the parameter. Subsequently, a reduction analogous to complete sufficiency is carried out, and within the complete E-sufficient space of inference functions, one is chosen with properties that we deem desirable.