Mixed Poisson distributions tail equivalent to their mixing distributions
type: Item
instance of: scholarly article
title: Mixed Poisson distributions tail equivalent to their mixing distributions
zbMATH Open document ID: 0917.60038
DOI: 10.1016/S0167-7152(98)00019-4
author: Richard Perline
published in: Statistics \& Probability Letters
publication date: 1999-08-02T00:00:00Z
review text: A mixed Poisson distribution is generated by the integral of Poisson mass function multiplied by the mixing density function of a nonnegative random variable. For most mixing density functions the integral to calculate mixed Poisson distribution is very complicated and cannot be computed in closed form but they give rise to attractive models for many applications [\textit{M. G. Bulmer}, Theor. Population Biology 5, 187-191 (1974), Biometrics 30, 101-110 (1974); \textit{S. A. Shaban}, Tamkang J. Math. 19, No. 1, 17-27 (1988; Zbl 0672.62099)]. But fortunately a mixed Poisson distribution can be tail equivalent to the mixing distribution in the sense of \textit{S. I. Resnick} [J. Appl. Probab. 8, 135-156 (1971; Zbl 0217.49903)]. The author identifies two broad classes of mixing distribution with this property: unbounded support with twice continuously differentiable mixing density functions for large \(t\) and that satisfy either: (a) the first von Mises condition, or (b) the third von Mises condition for domain of attraction of extreme value distribution and have hazard rates equal to \(o(t^{-\sigma})\) as \(t\to\infty\) for \(\sigma\geq {1\over 2}\).
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60G70
zbMATH DE Number: 1202795
zbMATH Keywords: mixed Poisson distribution
zbMATH Keywords: tail equivalence
zbMATH Keywords: extreme value theory
zbMATH Keywords: Poisson-lognormal distribution
reviewed by: Adhir K. Basu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1202795