On a Characterization of Zero-Inflated Negative Binomial Distribution ()
1. Introduction
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
(1)
where
, 
, 
, and
.
The probability generating function of X is given by
(2)
Hence the first derivative of 
is given by
2. Characterization
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
(3)
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
(4)
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
![]()
(5)
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
![]()
(6)
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] .