Displaced Poisson distribution

In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution.

Displaced Poisson Distribution
Probability mass function Displaced Poisson distributions for several values of ${\displaystyle \lambda }$ and ${\displaystyle r}$. At ${\displaystyle r=0}$, the Poisson distribution is recovered. The probability mass function is only defined at integer values.
Parameters	${\displaystyle \lambda \in (0,\infty )}$, ${\displaystyle r\in (-\infty ,\infty )}$
Support	${\displaystyle k\in \mathbb {N} _{0}}$
Mean	${\displaystyle \lambda -r}$
Mode	${\displaystyle {\begin{cases}\left\lceil \lambda -r\right\rceil -1,\left\lfloor \lambda -r\right\rfloor &{\text{if }}\lambda \geq r+1\\0&{\text{if }}\lambda <r+1\\\end{cases}}}$
Variance	${\displaystyle \lambda }$
MGF	${\displaystyle e^{\lambda \left(e^{t-1}\right)-tr}\cdot {\dfrac {I\left(r+s,\lambda e^{t}\right)}{I\left(r+s,\lambda \right)}}}$,  ${\displaystyle I\left(r,\lambda \right)=\sum _{y=r}^{\infty }{\dfrac {e^{-\lambda }\lambda ^{y}}{y!}}}$ When ${\displaystyle r}$ is a negative integer, this becomes ${\displaystyle e^{\lambda \left(e^{t-1}\right)-tr}}$

Definitions

Probability mass function

The probability mass function is

${\displaystyle P(X=n)={\begin{cases}e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r,\lambda \right)}},\quad n=0,1,2,\ldots &{\text{if }}r\geq 0\\[10pt]e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r+s,\lambda \right)}},\quad n=s,s+1,s+2,\ldots &{\text{otherwise}}\end{cases}}}$

where ${\displaystyle \lambda >0}$ and r is a new parameter; the Poisson distribution is recovered at r = 0. Here ${\displaystyle I\left(r,\lambda \right)}$ is the Pearson's incomplete gamma function:

${\displaystyle I(r,\lambda )=\sum _{y=r}^{\infty }{\frac {e^{-\lambda }\lambda ^{y}}{y!}},}$

where s is the integral part of r. The motivation given by Staff^[1] is that the ratio of successive probabilities in the Poisson distribution (that is ${\displaystyle P(X=n)/P(X=n-1)}$) is given by ${\displaystyle \lambda /n}$ for ${\displaystyle n>0}$ and the displaced Poisson generalizes this ratio to ${\displaystyle \lambda /\left(n+r\right)}$.

Examples

One of the limitations of the Poisson distribution is that it assumes equidispersion – the mean and variance of the variable are equal.^[2] The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as:

• the distribution of insect populations in crop fields;^[3]
• the number of flowers on plants;^[1]
• motor vehicle crash counts;^[4] and
• word or sentence lengths in writing.^[5]

Properties

Descriptive Statistics

• For a displaced Poisson-distributed random variable, the mean is equal to ${\displaystyle \lambda -r}$ and the variance is equal to ${\displaystyle \lambda }$.
• The mode of a displaced Poisson-distributed random variable are the integer values bounded by ${\displaystyle \lambda -r-1}$ and ${\displaystyle \lambda -r}$ when ${\displaystyle \lambda \geq r+1}$. When ${\displaystyle \lambda <r+1}$, there is a single mode at ${\displaystyle x=0}$.
• The first cumulant ${\displaystyle \kappa _{1}}$ is equal to ${\displaystyle \lambda -r}$ and all subsequent cumulants ${\displaystyle \kappa _{n},n\geq 2}$ are equal to ${\displaystyle \lambda }$.