# The COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

## TL;DR

This paper introduces the COM-negative binomial distribution, a flexible model for overdispersed and zero-inflated count data, with theoretical properties and practical applications demonstrated through parameter estimation and goodness-of-fit tests.

## Contribution

It develops a new three-parameter COM-type negative binomial distribution with comprehensive theoretical properties and applies it to real-world overdispersed, zero-inflated datasets.

## Key findings

- Distribution models overdispersion and zero-inflation effectively.
- Maximum likelihood estimation achieves good parameter fit.
- Distribution shows strong goodness-of-fit in applied datasets.

## Abstract

In this paper, we focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type $(a,b,0)$ class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein's identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.