The Space of Inference Functions: Ancillarity, Sufficiency and Projection

Abstract

In this chapter, we construct the space of inference functions and information theoretic notions of E-sufficiency and E-ancillarity within this space. Let X be a sample space, and P be a class of probability measures P on X. For each PεP we let V_P be the vector space of real valued functions f defined on the sample space X such that Ep[f(X)]² < ∞. We introduce the usual inner product defined on V_P,

$$<f_\textup{1},f_\textup{2}>_P=E_P\{f_\textup{1}(X)f_\textup{2}(X)\}$$

Let θ be a real valued function on the class of probability measures P and define the parameter space θ = {θ(P); PεP}. Note that θ need not be a one to one functional. If it is, we call the model a one-parameter model.