Displaced Poisson distribution explained
Displaced Poisson Distribution |
Type: | mass |
Pdf Caption: | Displaced Poisson distributions for several values of
and
. At
, the Poisson distribution is recovered. The probability mass function is only defined at integer values. |
Parameters: |
,
|
Support: |
|
Mean: |
|
Mode: | \begin{cases}
\left\lceilλ-r\right\rceil-1,\left\lfloorλ-r\right\rfloor&ifλ\geqr+1\\
0&ifλ<r+1\\
\end{cases}
|
Variance: |
|
Mgf: |
⋅ \dfrac{I\left(r+s,λet\right)}{I\left(r+s,λ\right)}
, I\left(r,λ\right)=
\dfrac{e-λλy}{y!}
When
is a negative integer, this becomes
|
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{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
and
r is a new parameter; the Poisson distribution is recovered at
r = 0. Here
is the Pearson's
incomplete gamma function:
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
) is given by
for
and the displaced Poisson generalizes this ratio to
.
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
and the variance is equal to
.
- The mode of a displaced Poisson-distributed random variable are the integer values bounded by
and
when
. When
, there is a single mode at
.
is equal to
and all subsequent cumulants
are equal to
.
Notes and References
- Staff. P. J. . The displaced Poisson distribution. Journal of the American Statistical Association. 1967. 62. 318. 643 - 654. 10.1080/01621459.1967.10482938.
- 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 .
- 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 .
- 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.