Mixed Poisson distributions tail equivalent to their mixing distributions
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 \(\sigma\geq {1\over 2}\).
scientific article; zbMATH DE number 1202795
scientific article