Abstract inference / Ulf Grenander

Bibliogr. p. 511-521. Index

This monograph provides an overview of inference methods developed for abstract sample spaces (like observed stochastic processes) as well as a description of the theory of estimation problems for abstract (i.e. infinite-dimensional) parameter spaces. It is divided into three parts.
Part I contains a collection of results from the theory of infinite-dimensional probability spaces related to inference problems. These results are well motivated by means of examples and formulated as simply as possible, which makes this part very readable.
Part II of the book contains general results on likelihood ratios, consistent estimation and linear inference (B.L.U.E. estimators) for mean value functions of stationary stochastic processes. There are numerous interesting applications for processes with continuous or discrete time (under various spectral conditions) and for random elements of several Banach spaces.
There is a separate chapter on testing of Gaussian processes as well as for test problems for the infinitesimal generators of finite Markov chains, birth and death processes, diffusion processes and processes with independent increments.
Part III describes inference in abstract parameter spaces containing yet unpublished results. The classical principles like maximum likelihood are applied to a sequence of regularized models or sieves approaching the given model (with increasing number of observations). This can be done in numerous ways dictated not only by consistency but also by computability. Very different examples are investigated and Chapter 9 contains a recent general formulation of the concept of sieves.
The book requires a solid background in probability (including infinite spaces), some statistics, real and complex analysis on the level of Riesz-Sz.-Nagy, and some knowledge of differential equations at the graduate student level.
The theorems are formulated with simple conditions rather than extreme generality and well motivated by numerous examples. Some of them are interesting on their own and invite further investigation. Sometimes the book is difficult to use as a quick reference for particular results because the notations used in these results are scattered over several chapters. Undoubtedly this book is a distinctive and very useful reference in its field. (MathSciNet)