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,infty)

,

r\in(-infty,infty)

Support:

k\inN0

Mean:

λ-r

Mode:

\begin{cases} \left\lceilλ-r\right\rceil-1,\left\lfloorλ-r\right\rfloor&ifλ\geqr+1\\ 0&ifλ<r+1\\ \end{cases}

Variance:

λ

Mgf:
λ\left(et-1\right)-tr
e

\dfrac{I\left(r+s,λet\right)}{I\left(r+s,λ\right)}

,  

I\left(r,λ\right)=

infty
\sum
y=r

\dfrac{eλy}{y!}

When

r

is a negative integer, this becomes
λ\left(et-1\right)-tr
e

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\dfrac{λn+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:
infty
I(r,λ)=\sum
y=r
eλy
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:

Properties

Descriptive Statistics

λ-r

and the variance is equal to

λ

.

λ-r-1

and

λ-r

when

λ\geqr+1

. When

λ<r+1

, there is a single mode at

x=0

.

\kappa1

is equal to

λ-r

and all subsequent cumulants

\kappan,n\geq2

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.