Introduction

The aim of this chapter is to construct a general geometrical setting, based on Hilbert space, in which one may study various estimation techniques, in particular with respect to efficiency and robustness. Given the sort of data one wishes to study, such as continuous, discrete, etc., the set of data-generation processes (DGPs) capable of generating data of that sort is given the structure of a Hilbert manifold. Statistical models will be treated as submanifolds of this underlying manifold.

Geometrical methods are frequently used in the study of statistical inference. One important strand of the literature is presented in Amari (1990), whose numerous earlier papers were inspired by some very abstract work of Chentsov (1972) and led to the concept of a statistical manifold. Other review papers and books in this tradition include Barndorff-Nielsen, Cox and Reid (1986), Kass (1989), Murray and Rice (1993), and Barndorff-Nielsen and Cox (1994). Most of this work makes use of finite-dimensional differential manifolds, which are usually representations of models in the exponential family.

Infinite-dimensional Hilbert space methods are extensively used in another strand of literature, for which the most suitable recent reference is Small and McLeish (1994). This book contains numerous references to the original papers on which it builds.