Displaced Poisson distribution

{{Infobox probability distribution

| name = Displaced Poisson Distribution

| type = mass

| pdf_image = 325px

| pdf_caption = 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 = $e^{\lambda \left( e^{t-1} \right) - tr } \cdot \dfrac{I \left( r+s, \lambda e^{t} \right)}{I \left( r+s, \lambda \right)}$ ,

$I \left(r, \lambda \right) = \sum^\infty_{y=r} \dfrac{e^{-\lambda} \lambda^y}{y!}$

When $r$ is a negative integer, this becomes $e^{\lambda \left( e^{t-1} \right) - tr }$

}}

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

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

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:

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

where s is the integral part of r.

The motivation given by Staff{{cite journal| last=Staff| first=P. J. | title=The displaced Poisson distribution| journal=Journal of the American Statistical Association| year=1967| volume=62| issue=318| pages=643–654| doi=10.1080/01621459.1967.10482938}} 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.{{Cite journal |last1=Chakraborty |first1=Subrata |last2=Ong |first2=S. H. |date=2017 |title=Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution |journal=Journal of Statistical Distributions and Applications |language=en |volume=4 |issue=1 |doi=10.1186/s40488-017-0060-9 |issn=2195-5832 |doi-access=free |arxiv=1411.0980 }} The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as:

the distribution of insect populations in crop fields;{{Cite journal |last=Staff |first=P. J. |date=1964 |title=The Displaced Poisson Distribution |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.1964.tb00146.x |journal=Australian Journal of Statistics |language=en |volume=6 |issue=1 |pages=12–20 |doi=10.1111/j.1467-842X.1964.tb00146.x |hdl=1959.4/66103 |issn=0004-9581|hdl-access=free }}
the number of flowers on plants;
motor vehicle crash counts;{{Cite journal |last1=Khazraee |first1=S. Hadi |last2=Sáez‐Castillo |first2=Antonio Jose |last3=Geedipally |first3=Srinivas Reddy |last4=Lord |first4=Dominique |date=2015 |title=Application of the Hyper‐Poisson Generalized Linear Model for Analyzing Motor Vehicle Crashes |url=https://onlinelibrary.wiley.com/doi/10.1111/risa.12296 |journal=Risk Analysis |language=en |volume=35 |issue=5 |pages=919–930 |doi=10.1111/risa.12296 |pmid=25385093 |bibcode=2015RiskA..35..919K |s2cid=206295555 |issn=0272-4332|url-access=subscription }} and
word or sentence lengths in writing.{{Citation |last1=Antić |first1=Gordana |title=Word Length and Frequency Distributions in Different Text Genres |date=2006 |url=http://link.springer.com/10.1007/3-540-31314-1_37 |work=From Data and Information Analysis to Knowledge Engineering |pages=310–317 |editor-last=Spiliopoulou |editor-first=Myra |access-date=2023-12-07 |place=Berlin/Heidelberg |publisher=Springer-Verlag |language=en |doi=10.1007/3-540-31314-1_37 |isbn=978-3-540-31313-7 |last2=Stadlober |first2=Ernst |last3=Grzybek |first3=Peter |last4=Kelih |first4=Emmerich |editor2-last=Kruse |editor2-first=Rudolf |editor3-last=Borgelt |editor3-first=Christian |editor4-last=Nürnberger |editor4-first=Andreas|url-access=subscription }}

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 $\lambda$ .

References

Category:Discrete distributions