Displaced Poisson distribution

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

Definitions

Probability mass function

The probability mass function is

$P(X=n)=\{\begin{array}{cc}{e}^{-\lambda }\frac{{\lambda }^{n+r}}{(n+r)!}\cdot \frac{1}{I(r,\lambda )},n=0,1,2,\dots & \text{if }r\ge 0\\ e^{-\lambda }\frac{{\lambda }^{n+r}}{(n+r)!}\cdot \frac{1}{I(r+s,\lambda )},n=s,s+1,s+2,\dots & \text{otherwise}\end{array}$

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

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

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 $\lambda /(n+r)$.

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 $\lambda \ge r+1$. 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 ${\kappa }_n,n\ge 2$ are equal to $\lambda$.