Construction and Extension of Dispersion Models

Abstract

There are two main classes of dispersion models studied in the literature: proper dispersion models (PDMs) and exponential dispersion models (EDMs). Models that are neither PDMs nor EDMs are termed here non-standard dispersion models (NSDMs). We develop techniques for constructing new PDMs and NSDMs, addressing an open problem concerning the construction and extension of NSDMs. Given a unit deviance, a dispersion model is obtained by solving a normalising integral equation. We use characteristic functions of symmetric absolutely continuous probability measures to construct large families of unit deviances for which this equation has a constant solution, yielding PDMs. We then derive non-constant normalising functions from spectral properties of the associated exponential kernel: isolated zeros of its Fourier transform give bounded perturbations, while zero sets of positive Lebesgue measure yield non-trivial $L^2(\mathbb{R})$ orthogonal perturbations and infinite-dimensional families. The resulting models are NSDMs. The constructions show that the classes of PDMs and NSDMs are vast; in particular, the PDM construction contains an infinite-dimensional family parametrised by symmetric absolutely continuous distributions, and explicit continuum-sized families of NSDMs are obtained.

arXiv comment29 pages, 3 figures. Version 3 repairs the constructions of PDMs and NSDMs. The characteristic-function construction of PDMs is reformulated using coercive unit deviances, and the former main argument is replaced by Fourier-based constructions using spectral zeros and $L^2$-orthogonality. Examples and exposition were revised accordingly

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