Displaced Poisson distribution

Displaced Poisson Distribution

Probability mass function Displaced Poisson distributions for several values of [math]\displaystyle{ \lambda }[/math] and [math]\displaystyle{ r }[/math]. At [math]\displaystyle{ r=0 }[/math], the Poisson distribution is recovered. The probability mass function is only defined at integer values.
Parameters	[math]\displaystyle{ \lambda\in (0, \infty) }[/math], [math]\displaystyle{ r\in (-\infty, \infty) }[/math]
Support	[math]\displaystyle{ k \in \mathbb{N}_0 }[/math]
Mean	[math]\displaystyle{ \lambda - r }[/math]
Mode	[math]\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 \lt r+1\\ \end{cases} }[/math]
Variance	[math]\displaystyle{ \lambda }[/math]
MGF	[math]\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)} }[/math],  [math]\displaystyle{ I \left(r, \lambda \right) = \sum^\infty_{y=r} \dfrac{e^{-\lambda} \lambda^y}{y!} }[/math] When [math]\displaystyle{ r }[/math] is a negative integer, this becomes [math]\displaystyle{ e^{\lambda \left( e^{t-1} \right) - tr } }[/math]

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

[math]\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} }[/math]

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

[math]\displaystyle{ I(r,\lambda)=\sum^\infty_{y=r}\frac{e^{-\lambda} \lambda^y}{y!}, }[/math]

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 [math]\displaystyle{ P(X=n)/P(X=n-1) }[/math]) is given by [math]\displaystyle{ \lambda/n }[/math] for [math]\displaystyle{ n\gt 0 }[/math] and the displaced Poisson generalizes this ratio to [math]\displaystyle{ \lambda/\left(n+r\right) }[/math].

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