A flexible and computationally tractable discrete distribution derived from a stationary renewal process

TL;DR

This paper introduces a new class of discrete distributions derived from stationary renewal processes, enabling simple mean modeling and closed-form probability calculations, with a focus on gamma and inverse Gaussian-based distributions.

Contribution

It presents a novel, flexible discrete distribution framework from renewal processes, with explicit mean functions and closed-form probabilities, especially emphasizing the gamma-based distribution.

Findings

Gamma-based distribution has attractive properties

Distribution allows regression of mean on covariates

Closed-form probability calculations are feasible

Abstract

A class of discrete distributions can be derived from stationary renewal processes. They have the useful property that the mean is a simple function of the model parameters. Thus regressions of the distribution mean on covariates can be carried out and marginal effects of covariates calculated. Probabilities can be easily computed in closed form for only two such distributions, when the event interarrival times in the renewal process follow either a gamma or an inverse Gaussian distribution. The gamma-based distribution has more attractive properties and is described and fitted to data. The inverse-Gaussian based distribution is also briefly discussed.