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

Definitions

Probability mass function

The probability mass function is

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

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

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

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

Examples

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

• the distribution of insect populations in crop fields;
• the number of flowers on plants;
• motor vehicle crash counts; and
• word or sentence lengths in writing.

Properties

Descriptive Statistics

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