On the Maximum and Minimum of a Multivariate Poisson Distribution

TL;DR

This paper derives explicit cumulative distribution functions for the maximum and minimum of multivariate Poisson distributions under three dependence structures, revealing significant differences and offering new insights into dependence modeling.

Contribution

It introduces explicit formulas for the CDFs of multivariate Poisson maxima and minima under three dependence models, including a new thinning-dependent distribution.

Findings

Significant differences between dependence structures in multivariate Poisson distributions

Explicit CDF formulas for maximum and minimum under each dependence model

Numerical examples illustrating the theoretical results

Abstract

In this paper, we investigate the cumulative distribution functions (CDFs) of the maximum and minimum of multivariate Poisson distributions with three dependence structures, namely, the common shock, comonotonic shock and thinning-dependence models. In particular, we formulate the definition of a thinning-dependent multivariate Poisson distribution based on Wang and Yuen (2005). We derive explicit CDFs of the maximum and minimum of the multivariate Poisson random vectors and conduct asymptotic analyses on them. Our results reveal the substantial difference between the three dependence structures for multivariate Poisson distribution and may suggest an alternative method for studying the dependence for other multivariate distributions. We further provide numerical examples demonstrating obtained results.