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Research output: Working paper › Research

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. This paper exposes a technique for constructing new proper dispersion models and non-standard dispersion models. 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. It turns that 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 (\ie, 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. On the one hand, the dispersion models constructed with constant normalising functions (corresponding to the trivial solution of the integral equation) are all proper dispersion models; on the other hand, the normalising functions arising from non-trivial solutions of the integral equation generate dispersion models that are non-standard models. As a consequence, 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.

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. It turns that 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 (\ie, 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. On the one hand, the dispersion models constructed with constant normalising functions (corresponding to the trivial solution of the integral equation) are all proper dispersion models; on the other hand, the normalising functions arising from non-trivial solutions of the integral equation generate dispersion models that are non-standard models. As a consequence, 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.

Original language | English |
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Publisher | arXiv |

Number of pages | 19 |

Publication status | Published - Aug 2020 |

- Dispersion Models, Exponential Dispersion Models, Proper Dispersion Models, Generalised Functions, Non Standard Dispersion Models

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ID: 196436130