On a Characterization of Zero-Inflated Negative Binomial Distribution

Abstract

Zero-inflated negative binomial distribution is characterized in this paper through a linear differential equation satisfied by its probability generating function.

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Suresh, R. , Nanjundan, G. , Nagesh, S. and Pasha, S. (2015) On a Characterization of Zero-Inflated Negative Binomial Distribution. Open Journal of Statistics, 5, 511-513. doi: 10.4236/ojs.2015.56053.

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] .

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] 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 [2] Nanjundan, G. and Pasha, S. (2015) A Note on the characterization of Zero-Inflated Poisson Distribution. Open Journal of Statistics, 5, 140-142.http://dx.doi.org/10.4236/ojs.2015.52017 [3] Nagesh, S., Nanjundan, G., Suresh, R. and Pasha, S. (2015) A Characterization of Zero-Inflated Geometric Model. International Journal of Mathematics Trends and Technology, 23, 71-73. http://dx.doi.org/10.14445/22315373/IJMTT-V23P510