Zero-inflated negative binomial distribution is characterized in this paper through a linear differential equation satisfied by its probability generating function.
Zero-inflated discrete distributions have paved ways for a wide variety of applications, especially count re- gression models. Nanjundan [
A random variable X is said to have a zero-inflated negative binomial distribution, if its probability mass function is given by
where
The probability generating function of X is given by
Hence the first derivative of
The following theorem characterizes the zero-inflated negative binomial distribution.
Theorem 1 Let X be a non-negative integer valued random variable with
where a, b, c are constants.
Proof. 1) Suppose that X has zero-inflated negative binomial distribution with the probability mass function specified in (1). Then its pgf can be expressed as
Hence
2) Suppose that the pgf of x satisfies the linear differential equation in (3).
Writing the Equation (3) as
we get
On integrating both sides w.r.t. x, we get
That is
The solution of the differential equation in (3) becomes
If either b or c or both are equal to zero, then
are non-zero. Since
negative coefficients, which is not permissible because
where n is a positive integer. Since
Therefore
Hence
This completes the proof of the theorem.
Also, it can be noted that when
R.Suresh,G.Nanjundan,S.Nagesh,SadiqPasha, (2015) On a Characterization of Zero-Inflated Negative Binomial Distribution. Open Journal of Statistics,05,511-513. doi: 10.4236/ojs.2015.56053