Zero-inflated discrete distributions have paved ways for a wide variety of applications, especially count re- gression models. Nanjundan [1] has characterized a subfamily of power series distributions whose probability generating function (pgf) satisfies the differential equation, where is the first derivative of. This subfamily includes binomial, Poisson, and negative binomial distributions. Also, Nanjundan and Sadiq Pasha [2] have characterized zero-inflated Poisson distribution through a differential equation. In the similar way, Nagesh et al. [3] have characterized zero-inflated geometric distribution. Along the same lines, zero-inflated negative binomial distribution is characterized in this paper via a differential equation satisfied by its pgf.

A random variable X is said to have a zero-inflated negative binomial distribution, if its probability mass function is given by

where, , , and.

The probability generating function of X is given by

Hence the first derivative of is given by

The following theorem characterizes the zero-inflated negative binomial distribution.

Theorem 1 Let X be a non-negative integer valued random variable with. Then X has a zero-inflated negative binomial distribution if and only if its pgf satisfies

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 in (4) satisfies (3) with.

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 and hence has no meaning. Thus, both b and c

are non-zero. Since is a pgf, it is a power series of the type. When either or

is not a negative integer, the expansion of the factor on the right hand side of (5) will have

negative coefficients, which is not permissible because is a pgf. Hence the equation in (5) can be written as

where n is a positive integer. Since,.

Therefore

Hence in (6) satisfies (2) with, , , and.

This completes the proof of the theorem.

Also, it can be noted that when, the negative binomial distribution reduces to geometric distri- bution and the Theorem 1 in Section 2 concurs with the characterization result of Nagesh et al. [3] .

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