Pivotal measures in statistical experiments and sufficiency (Q1320431)

The purpose of this monograph is to develop a theory of sufficiency general enough to include situations that are not covered by earlier theories. An experiment \(E\) is a triplet \((X,A,P)\) consisting of a set \(X\), a \(\sigma\)-field \(A\) of subsets of \(X\), and a family \(P\) of probability measures on \((X,A)\). The members of \(P\) are indexed by a parameter \(\theta \in \Theta^*\). A dominated experiment has a dominating measure which is \(\sigma\)-finite. If we drop the \(\sigma\)- finiteness assumption, we have an undominated experiment. For any measure \(m\) on \((X,A)\), \(N(m)\) denotes the family of sets which are assigned measure zero by \(m\). An experiment \(E\) is said to be majorized if there exists a measure \(m\) on \((X,A)\) such that each \(P_ \theta\) in \(P\) has a density \(dP_ \theta/dm\) with respect to \(m\). Such an \(m\) is called a majorizing measure for \(E\). If \(N(P) = N(m)\), then \(m\) is called an equivalent majorizing measure. A subfield \(B\) is said to be sufficient for \(E\) if, for any \(a\in A\), there exists a \(B\)-measurable function \(E(I_ a | B)\), such that \[ P_ \theta (a \cap b) = \int_ b E(I_ a | B) (x)dP_ \theta \quad\text{for all } b \in B \text{ and } \theta \in \Theta^*. \] If \(B\) is sufficient for any pair from \(P\) then \(B\) is called pairwise sufficient for \(E\). If \(E\) is a majorized experiment, a subfield \(B\) is said to be PSS (pairwise sufficient with support) for \(E\) if it is pairwise sufficient and there exists a support \(S(\theta)\) of \(P_ \theta\) belonging to \(B\) for each \(\theta \in \Theta^*\). An equivalent majorizing measure \(n\) for a majorized experiment \(E\) is called a pivotal measure for \(E\) if the following condition is satisfied: for any subfield \(B\) of \(A\), \(B\) is PSS if and only if there exists a \(B\)-measurable density of \(P_ \theta\) with respect to \(n\) for each \(\theta \in \Theta^*\). It is shown that for any majorized experiment there exists a pivotal measure. PSS is a concept which is weaker than sufficiency and stronger than pairwise sufficiency. In the dominated case all three concepts coincide. In undominated cases, sufficiency has certain difficulties. In particular, a minimal sufficient subfield may not exist, and a nonsufficient subfield may include a sufficient one. The more general theory built on the concepts of pivotal measure and PSS avoids these difficulties. The chapter headings are: Introduction; PSS, pivotal measure and Neyman factorization; Structure of pairwise sufficient subfields and PSS; The Rao-Blackwell theorem and UMVUE; Common conditional probability for PSS and its applications; Structure of pivotal measure.