Mixed Poisson process with Max-U-Exp mixing variable -- Working version

TL;DR

This paper introduces the Max-U-Exp distribution, explores its properties within a Mixed Poisson process framework, and derives related distributions and their characteristics, providing new insights into stochastic modeling with this distribution.

Contribution

It defines and investigates the Max-U-Exp distribution and its role in a new class of Mixed Poisson processes, extending existing theories with novel distributional properties.

Findings

Max-U-Exp distribution properties thoroughly analyzed

Distribution of inter-arrival times characterized as Erlang-Max-U-Exp

Finite-dimensional and conditional distributions of the process derived

Abstract

This work defines and investigates the properties of the Max-U-Exp distribution. The method of moments is applied in order to estimate its parameters. Then, by using the previous general theory about Mixed Poisson processes, developed by Grandel (1997), and Karlis and Xekalaki (2005), and analogously to Jordanova et al. (2023), and Jordanova and Stehlik (2017) we define and investigate the properties of the new random vectors and random variables, which are related with this particular case of a Mixed Poisson process. Exp-Max-U-Exp distribution is defined and thoroughly investigated. It arises in a natural way as a distribution of the inter-arrival times in the Mixed Poisson process with Max-U-Exp mixing variable. The distribution of the renewal moments is called Erlang-Max-U-Exp and is defined via its probability density function. Investigation of its properties follows. Finally, the corresponding Mixed Poisson process with Max-U-Exp mixing variable is defined. Its finite dimensional and conditional distributions are found and their numerical characteristics are determined.