  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.
