Synopses & Reviews
The book discusses the estimation theory for the wide class of inhomogeneous Poisson processes. The consistency, limit distributions and the convergence of moments of parameter estimators are established in regular and non-regular (change-point type) problems. The maximum likelihood, Bayesian, and the minimum distance estimators are investigated in parametric problems and the empiric intensity measure and the kernel-type estimators are studied in nonparametric estimation problems. The properties of the estimators are also described in the situations when the observed Poisson process does not belong to the parametric family (no true model), when there are many true models (nonidentifiable family), when the observation window can be chosen by an optimal way, and others. The question of asymptotic efficiency of estimators is discussed in all of these problems. The book will be useful for those who use models of Poisson processes in their research. The large number of examples of inhomogeneous Poisson processes discussed in the book are taken from the fields of optical communications, reliability, image processing, and nuclear medicine. The material is suitable for graduate courses on stochastic processes. The book assumes familiarity with probability theory and mathematical statistics. Yury A. Kutoyants, Professor of Mathematics at the University of Main, Le Mans, France, is a member of the Bernoulli Society, the Mathematical Society of France, and the Institute of Mathematical Statistics. He is associate editor of "Finance and Stochastics" and "Statistical Inference for Stochastic Processes." He is author of "Parameter Estimation for Stochastic Processes" (Heldermann Verlag, Berlin, 1984) and "Identification of Dynamical Systems with Small Noise" (Kluwer, Dordrecht, 1994), and the of about 70 articles on the
Synopsis
This work is devoted to several problems of parametric (mainly) and nonparametric estimation through the observation of Poisson processes defined on general spaces. Poisson processes are quite popular in applied research and therefore they attract the attention of many statisticians. There are a lot of good books on point processes and many of them contain chapters devoted to statistical inference for general and partic- ular models of processes. There are even chapters on statistical estimation problems for inhomogeneous Poisson processes in asymptotic statements. Nevertheless it seems that the asymptotic theory of estimation for nonlinear models of Poisson processes needs some development. Here nonlinear means the models of inhomogeneous Pois- son processes with intensity function nonlinearly depending on unknown parameters. In such situations the estimators usually cannot be written in exact form and are given as solutions of some equations. However the models can be quite fruitful in en- gineering problems and the existing computing algorithms are sufficiently powerful to calculate these estimators. Therefore the properties of estimators can be interesting too.
Synopsis
The book represents highly theoretical investigations in statistical estimation for spatial Poisson processes. It uses a deep technique developed in the works by Hajek, Le Cam, Ibragimov, Khas'minskii, and the author himself. The asymptotic behavior of these estimators is investigated in detail for both regular and singular cases.
Description
Includes bibliographical references (p. 265-274) and index.
Table of Contents
Auxiliary Results.- First Properties of Estimators.- Asymptotic Expansions.- Nonstandard Problems.- The Change-poing Problems.- Nonparametric Estimation.