Mixed Poisson process with Stacy mixing variable

Abstract

Stacy distribution defined for the first time in 1961 provides a flexible framework for modeling of a wide range of real-life behaviors. It appears under different names in the scientific literature and contains many useful particular cases. Homogeneous Poisson processes are appropriate apriori models for the number of renewals up to a given time t > 0. This article mixes them and considers a mixed Poisson process with Stacy mixing variable. We call it a Poisson-Stacy process. The resulting counting process is one of the Generalized Negative Binomial processes, and the distribution of its time-intersections are very-well investigated in the scientific literature. Here we define and investigate their joint probability distributions. Then, the corresponding mixed renewal process is investigated and Exp-Stacy and Erlang-Stacy distributions are defined and partially studied. The article finishes with a simulation study of these stochastic processes. Some plots of the probability density functions, probability mass functions, mean square regressions, and sample paths are drawn together with the corresponding code for the simulations.

KEYWORDS:

• Mixed Poisson processes
• Stacy distribution
• Kratzel inegral

Disclosure statement

The authors report there are no competing interests to declare.

Notes

1 If a r.v. ξ has some probability law, we call l-type the set of distributions of all r.vs. $a\xi +b$, where $a\in \mathbb{R}$ and $b\in \mathbb{R}$. l comes from “linear”.

Additional information

Funding

The work was supported by the Scientific Research Fund in Konstantin Preslavsky University of Shumen, Bulgaria under Grant RD-08-35/18.01.2023; and ANID Chile – Proyecto Exploracion ANID 2022, No. 13220184.