A Functional Analytic Approach to Statistical Experiments: (236 Pitman Research Notes in Mathematics Series)

This research note deals with a structural theory of statistical experiments irrespective of the complexity of the underlying model. To avoid involved measure theoretic arguments, functional analytic means are employed, following LeCam's decision theoretic approach. The book provides a self-contained and unified introduction to the concept of generalized random variables and also includes a new, systematic discussion of the multivariate and even infinite-dimensional case. After the concepts of loss functions, subfields, and conditional expectation in this context have been presented, emphasis is laid on sufficiency, invariance, as well as optimality and admissibility in unbiased estimation. An appendix is devoted to an alternative approach due to Torgersen. Apart from original results, this work also contains new proofs and characterizations of facts occurring in the periodic literature only, faciliating access also to researchers outside the fields of this branch of statistical decision theory and/or advanced measure theory.

Table of Contents:
Part 1 Generalized random elements and convex operations: experiments and their L- and M-spaces; convex operations and multiplication; generalized random elements taking values in a Banach space; convex loss functions on Banach spaces; representation of generalized random elements. Part 2 Sub-algebras and conditional expectations: representations of sub-algebras in M; projections and conditional expectations; Jensen's inequality. Part 3 Sufficient subalgebras and projections; sufficiency characterizations and related properties; intersections, the minimal sufficient sub-algebra; Banach space results. Part 4 Optimality and admissibility in unbiased estimation: unbiased estimation in the "classical" context - a summary; auxiliary results; optimality and admissibility; the role of sufficiency and completeness; existence of nontrivial optimal estimators; generalized Banach space valued unbiased estimation. Part 5 Invariance: transitions; sufficiency and invariance, full models; calculation of IIG, I - special group properties; calculations of IIG, II - invariance approximation property; maximal invariants and metric transitivity; Banach space results. Appendix: alternative representation of M-spaces.