The limiting behavior of some infinitely divisible exponential dispersion models

Shaul K. Bar-Lev and Gérard Letac

Statistics & Probability Letters, 2010, vol. 80, issue 23-24, 1870-1874

Abstract: Consider an exponential dispersion model (EDM) generated by a probability [mu] on [0,[infinity]) which is infinitely divisible with an unbounded Lévy measure [nu]. The Jørgensen set (i.e., the dispersion parameter space) is then , in which case the EDM is characterized by two parameters: [theta]0, the natural parameter of the associated natural exponential family, and the Jørgensen (or dispersion) parameter, t. Denote the corresponding distribution by and let Yt be a r.v. with distribution . Then for [nu]((x,[infinity]))~-llogx around zero, we prove that the limiting law F0 of as t-->0 is a Pareto type law (not depending on [theta]0) with the form F0(u)=0 for u =1. This result enables an approximation of the distribution of Yt to be found for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.

Keywords: Exponential; dispersion; model; Infinitely; divisible; distributions; Limiting; distributions; Natural; exponential; family (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (1)

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