^{1}

^{*}

^{2}

^{*}

^{1}

^{*}

^{2}

^{*}

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