Displaced Poisson distribution explained

Displaced Poisson Distribution
Type:	mass
Pdf Caption:	Displaced Poisson distributions for several values of λ and r . At r =0 , the Poisson distribution is recovered. The probability mass function is only defined at integer values.
Parameters:	λ \in ( 0 , inf ty) , r\in ( -inf ty, inf ty)
Support:	k \in N _{0}
Mean:	λ - r
Mode:	\begin{cases} \left\lceil λ - r \right\rceil - 1 , \left\lfloor λ - r \right\rfloor & if λ \geq r +1 \\ 0 & if λ < r +1 \\ \end{cases}
Variance:	λ
Mgf:	e ^{λ \left( e ^{t-1} \right) - tr} ⋅ \dfrac{I \left( r + s, λ e ^{t} \right)}{I \left( r + s, λ \right)} ,  I \left(r, λ \right) = \sum ^{inf ty} _{y = r} \dfrac{e ^{-λ} λ ^{y}{y!}} When r is a negative integer, this becomes e ^{λ \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 ^{-λ} \dfrac{ λ ^{n+r} }\cdot\dfrac, \quad n=0,1,2,\ldots &\text r\geq 0\\[10pt] e^\dfrac\cdot\dfrac,\quad n=s,s+1,s+2,\ldots &\text \end

where

λ > 0 and r is a new parameter; the Poisson distribution is recovered at r = 0. Here I\left(r, λ \right) is the Pearson's incomplete gamma function:I(r, λ ) = \sum ^{inf ty} _{y = r} (e ^{-λ} λ ^{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 P(X = n)/P(X = n -1 ) ) is given by λ /n for n> 0 and the displaced Poisson generalizes this ratio to λ /\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;
• motor vehicle crash counts;^[4] and
• word or sentence lengths in writing.

Properties

Descriptive Statistics

• For a displaced Poisson-distributed random variable, the mean is equal to

λ - r and the variance is equal to λ .

• The mode of a displaced Poisson-distributed random variable are the integer values bounded by

λ - r - 1 and λ - r when λ \geq r +1 . When λ < r +1 , there is a single mode at x =0 .\kappa _{1} is equal to λ - r and all subsequent cumulants \kappa _{n} , n \geq 2 are equal to λ .

Notes and References

1. Staff. P. J. . The displaced Poisson distribution. Journal of the American Statistical Association. 1967. 62. 318. 643 - 654. 10.1080/01621459.1967.10482938.
2. Chakraborty . Subrata . Ong . S. H. . 2017 . Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution . Journal of Statistical Distributions and Applications . en . 4 . 1 . 10.1186/s40488-017-0060-9 . 2195-5832 . free . 1411.0980 .
3. Staff . P. J. . 1964 . The Displaced Poisson Distribution . Australian Journal of Statistics . en . 6 . 1 . 12–20 . 10.1111/j.1467-842X.1964.tb00146.x . 1959.4/66103 . 0004-9581. free .
4. Khazraee . S. Hadi . Sáez‐Castillo . Antonio Jose . Geedipally . Srinivas Reddy . Lord . Dominique . 2015 . Application of the Hyper‐Poisson Generalized Linear Model for Analyzing Motor Vehicle Crashes . Risk Analysis . en . 35 . 5 . 919–930 . 10.1111/risa.12296 . 25385093 . 2015RiskA..35..919K . 206295555 . 0272-4332.