Characterizations and generalizations of the negative binomial distribution

Abstract

In this paper, we give detailed descriptions of the Zero-Modified Negative Binomial distribution for analyzing count data. In particular, we study the characterizations and properties of this distribution, whose main advantage is its flexibility which makes it suitable for modeling a wide range of overdispersed and underdispersed count data (which may or may not be caused by zero-modification, i.e., the inflation or deflation of zeroes), without requiring previous knowledge about any of these inherent data characteristics. We derive maximum likelihood estimation of the model parameters based on positive observations, and evaluate the loss of efficiency by considering this procedure. We illustrate the suitability of this distribution on real data sets with different types of zero-modification.

Fisher score method

We considered the Fisher score method to calculate the maximum likelihood estimates for $\mu$ and $\phi$ parameters of ZMNB (or ZTNB) distribution. For this, we use the iterative equations:

$$\begin{aligned} \left( \begin{array}{c} \mu ^{(j+1)}\\ \phi ^{(j+1)}\\ \end{array} \right) =\left( \begin{array}{c} \mu ^{(j)}\\ \phi ^{(j)}\\ \end{array} \right) +\left[ \begin{array}{ccccccccc} \mathcal {J}^{^+(j)}_{\mu \mu } &{} \mathcal {J}^{^+(j)}_{\mu \phi }\\ \mathcal {J}^{^+(j)}_{\phi \mu } &{} \mathcal {J}^{^+(j)}_{\phi \phi }\\ \end{array}\right] ^{-1}\times \left( \begin{array}{c} \mathcal {U}_{\mu }^{^+(j)}\\ \mathcal {U}_{\phi }^{^+(j)}\\ \end{array} \right) . \end{aligned}$$

The maximum likelihood estimates of the parameters are obtained when $(\mathcal {U}_{\mu }^{^+(j)})^2+(\mathcal {U}_{\phi }^{^+(j)})^2< \varepsilon$ occur, where $\varepsilon$ is the error in the estimation (ie, when the difference between iterations is less than a pre-established error $\varepsilon$). A detailed description of the Fisher score algorithm is presented as follows: