# Construction and Extension of Dispersion Models

There are two main classes of dispersion models studied in the literature: proper (PDM), and exponential dispersion models (EDM). Dispersion models that are neither proper nor exponential dispersion models are termed here non-standard dispersion models (NSDM). This paper exposes a technique for constructing new PDMs and NSDMs. This construction provides a solution to an open question in the theory of dispersion models about the extension of non-standard dispersion models. Given a unit deviance function, a dispersion model is usually constructed by calculating a normalising function that makes the density function integrates one. This calculation involves the solution of non-trivial integral equations. The main idea explored here is to use characteristic functions of real non-lattice symmetric probability measures to construct a family of unit deviances that are sufficiently regular to make the associated integral equations tractable. The integral equations associated to those unit deviances admit a trivial solution, in the sense that the normalising function is a constant function independent of the observed values. However, we show, using the machinery of distributions (i.e., generalised functions) and expansions of the normalising function with respect to specially constructed Riez systems, that those integral equations also admit infinitely many non-trivial solutions, generating many NSDMs. We conclude that, the cardinality of the class of non-standard dispersion models is larger than the cardinality of the class of real non-lattice symmetric probability measures.

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