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The theory of dispersion models. (English) Zbl 0928.62052

Monographs on Statistics and Applied Probability. 76. London: Chapman & Hall. xii, 237 p. (1997).
This book is an introduction to the theory of dispersion models. The main raison d’être for dispersion models is to serve as error distributions for generalized linear models. Generalized linear models are now fairly mainstream, and have inspired new developments in other areas such as for example longitudinal data and time series analysis. It hence seems timely to present a detailed study of dispersion models and their use in generalized linear models. The present volume deals with the theoretical aspects of dispersion models as such. A second volume is planned, on the statistical analysis for generalized linear models based on dispersion model error distributions.
The theory of dispersion models straddles both statistics and probability, and involves an encyclopaedic collection of tools, such as exponential families, asymptotic theory, stochastic processes, Tauber theory, infinite divisibility and stable distributions. In particular, the study of variance functions for natural exponential families involves a variety of tools from probability and analysis, and has branched out as a separate research topic.
The common theme that emerges from this diversity of methods and models is the use of the deviance and the variance function. These two items determine the main aspects of the shape of the distributions. The systematic use of these two functions outside exponential families provides a convenient unification of results for exponential and proper dispersion models. In generalized linear models, the deviance and the variance function play key roles in analysis of deviance and residual analysis, respectively. This allows a unified approach to distribution theory and statistical analysis for generalized linear models based on dispersion models.
Concerns about application of the theory to generalized linear models have shaped the exposition in various ways. I concentrate on the univariate case, which is the most important one from a practical point of view. I emphasize the interpretation of the distributions and the development of models for different data types, and emphasize models that are useful for statistical data analysis. In order to make the exposition more accessible to graduate students, I have included introductory material on moment generating functions and natural exponential families.
This approach should make the book useful as both a research reference and a graduate-level textbook in distribution theory for generalized linear models. Prerequisites include measure-theoretic probability and some background in statistical inference. However, measure theory is used mainly in the definitions of natural exponential families and exponential dispersion models, and most of the results are accessible without detailed knowledge of measure theory. (From the preface.)
Contents (Chapter headings): 1. Introduction to dispersion models. 2. Natural exponential families. 3. Exponential dispersion models. 4. Tweedie models. 5. Proper dispersion models.

MSC:

62J12 Generalized linear models (logistic models)
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62E20 Asymptotic distribution theory in statistics
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
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