Open Journal of Statistics
Vol.05 No.02(2015), Article ID:55814,2 pages
10.4236/ojs.2015.52017
A Note on the Characterization of Zero-Inflated Poisson Model
G. Nanjundan, Sadiq Pasha
Department of Statistics, Bangalore University, Bangalore, India
Email: nanzundan@gmail.com
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 14 March 2015; accepted 16 April 2015; published 20 April 2015
ABSTRACT
Zero-Inflated Poisson model has found a wide variety of applications in recent years in statistical analyses of count data, especially in count regression models. Zero-Inflated Poisson model is characterized in this paper through a linear differential equation satisfied by its probability generating function [1] [2] .
Keywords:
Zero-Inflated Poisson Model, Probability Generating Function, Linear Differential Equation
1. Introduction
A random variable X is said to have a zero-inflated Poisson distribution if its probability mass function is given by
(1)
where and
,
,
.
Thus, the distribution of X is a mixture of a distribution degenerate at zero and a Poisson distribution with mean.
2. Probability Generating Function
The probability generating function (pgf) of X is given by
.
3. Characterization
Let X be a non-negative integer valued random variable with and the pgf
. Then, the distribution of X is zero-inflated Poisson if and only if
, where
, b are constants and
is the first derivative of
.
Proof:
1) Suppose that X has a zero-inflated Poisson distribution specified in (1.1). Then the pgf of X is given by
On differentiation, we get
.
Hence satisfies the linear differential equation
(2)
2) Suppose that the pgf of X satisfies
If, then
and in turn
. By the property of the pgf,
. But
, which is not possible because
.
Therefore.
3) The Linear Differential Equation
The linear differential equation is of the form
where and
are functions of
.
Then its solution is given by
,
where c is an arbitrary constant.
Here
.
Hence,
.
Therefore the solution of the Equation (2) is given by
.
We now extract the probabilities,
using the above solution.
Since is a pgf,
, where
is the k-th derivative of
.
We get
,
,
, and so on.
Now,
Since, it is easy to see that
,
We have
with and
.
Therefore X has the pgf specified in Equation (1).
References
- Nanjundan, G. (2011) A Characterization of the Members of a Subfamily of Power Series Distributions. Applied mathematics, 2, 750-751. http://dx.doi.org/10.4236/am.2011.26099
- Nanjundan, G. and Ravindra Naika, T. (2012) An Asymptotic Comparison of the Maximum Likelihood and the Moment Estimators in a Zero-Inflated Poisson Model. Applied mathematics, 3, 610-617. http://dx.doi.org/10.4236/am.2012.36095