An introduction to Bent Jorgensen's ideas

TL;DR

This paper introduces the theory of dispersion models, highlighting their mathematical structure, new construction techniques, and applications in statistical modeling, expanding the understanding of their diversity and practical use.

Contribution

It presents a new characteristic function-based method for constructing dispersion models, broadening the class of models beyond traditional exponential and proper dispersion types.

Findings

Introduction of a new convolution-based construction technique

Demonstration of a large variety of dispersion models, including non-proper and non-exponential types

Applications to generalized linear models and dependent data models

Abstract

We briefly expose some key aspects of the theory and use of dispersion models, for which Bent Jorgensen played a crucial role as a driving force and an inspiration source. Starting with the general notion of dispersion models, built using minimalistic mathematical assumptions, we specialize in two classes of families of distributions with different statistical flavors: exponential dispersion and proper dispersion models. The construction of dispersion models involves the solution of integral equations that are, in general, untractable. These difficulties disappear when a more mathematical structure is assumed: it reduces to the calculation of a moment generating function or of a Riemann-Stieltjes integral for the exponential dispersion and the proper dispersion models, respectively. A new technique for constructing dispersion models based on characteristic functions is introduced turning the integral equations above into a tractable convolution equation and yielding examples of dispersion models that are neither proper dispersion nor exponential dispersion models. A corollary is that the cardinality of regular and non-regular dispersion models are both large. Some selected applications are discussed including exponential families non-linear models (for which generalized linear models are particular cases) and several models for clustered and dependent data based on a latent Levy process.