# A Closed Form Approximation of Moments of New Generalization of Negative Binomial Distribution

## TL;DR

This paper introduces a closed-form approximation for the mean and variance of a new, flexible generalization of the negative binomial distribution derived from the Extended COM-Poisson distribution, enhancing its practical utility.

## Contribution

It provides the first closed-form approximations for the moments of the NGNB distribution, facilitating its use in regression models and extended applications.

## Key findings

- Closed-form approximations for mean and variance of NGNB.
- Approximations relate to convolution of negative binomial distributions.
- Enhances applicability in generalized negative binomial regression.

## Abstract

In this paper, we propose a closed form approximation to the mean and variance of a new generalization of negative binomial (NGNB) distribution arising from the Extended COM-Poisson (ECOMP) distribution developed by Chakraborty and Imoto (2016)(see [4]). The NGNB is a special case of the ECOMP distribution and was named so by these authors. This distribution is more flexible in terms of the dispersion index as compared to its ordinary counterparts. It approaches the COM-Poisson distribution (Shmueli et al. 2005) [11] under suitable limiting conditions. The NGNB can also be obtained from the COM-Negative Hypergeometric distribution (Roy et al. 2019)[10] as a limiting distribution. In this paper, we present closed-form approximations for the mean and variance of the NGNB distribution. These approximations can be viewed as the mean and variance of convolution of independent and identically distributed negative binomial populations. The proposed closed-form approximations of the mean and variance will be helpful in building the link function for the generalized negative binomial regression model based on the NGNB distribution and other extended applications, hence resulting in enhanced applicability of this model.