Structure of the probability mass function of the Poisson distribution of order $k$

TL;DR

This paper characterizes the shape and parameter space of the Poisson distribution of order k, providing visual interpretation, analyzing its modes, and proposing new bounds and inequalities for its properties.

Contribution

It offers a detailed shape analysis of the pmf of the Poisson distribution of order k, including parameter space mapping and mode structure insights.

Findings

The pmf has a specific partitioned shape with peaks and a single zero point.

Parameter space boundaries determine the pmf's behavior and mode structure.

New bounds and conjectures for median and mode are proposed.

Abstract

The Poisson distribution of order $k$ is a special case of a compound Poisson distribution. For $k=1$ it is the standard Poisson distribution. Although its probability mass function (pmf) is known, what is lacking is a $visual$ interpretation, which a sum over terms with factorial denominators does not supply. Unlike the standard Poisson distribution, the Poisson distribution of order $k$ can display a maximum of $four$ peaks simultaneously, as a function of two parameters: the order $k$ and the rate parameter $\lambda$. This note characterizes the shape of the pmf of the Poisson distribution of order $k$. The pmf can be partitioned into a single point at $n=0$, an increasing sequence for $n \in [1,k]$ and a mountain range for $n>k$ (explained in the text). The ``parameter space'' of the pmf is mapped out and the significance of each domain is explained, in particular the change in behavior of the pmf as a domain boundary is crossed. A simple analogy (admittedly unrelated) is that of the discriminant of a quadratic with real coefficients: its domains characterize the nature of the roots (real or complex), and the domain boundary signifies the presence of a repeated root. Something similar happens with the pmf of the Poisson distribution of order $k$. As an application, this note explains the mode structure of the Poisson distribution of order $k$. Improvements to various inequalities are also derived (sharper bounds, etc.). New conjectured upper and lower bounds for the median and the mode are also proposed.