Bridging the Gap: A Generalized Stochastic Process for Count Data

ABSTRACT

The Bernoulli and Poisson processes are two popular discrete count processes; however, both rely on strict assumptions. We instead propose a generalized homogenous count process (which we name the Conway–Maxwell–Poisson or COM-Poisson process) that not only includes the Bernoulli and Poisson processes as special cases, but also serves as a flexible mechanism to describe count processes that approximate data with over- or under-dispersion. We introduce the process and an associated generalized waiting time distribution with several real-data applications to illustrate its flexibility for a variety of data structures. We consider model estimation under different scenarios of data availability, and assess performance through simulated and real datasets. This new generalized process will enable analysts to better model count processes where data dispersion exists in a more accommodating and flexible manner.

KEYWORDS:

• Bernoulli process
• Conway-Maxwell-Poisson (COM-Poisson) distribution
• Count process
• Dispersion
• Poisson process
• Waiting time

Acknowledgments

This article is released to inform interested parties of research and to encourage discussion. The views expressed are those of the authors and not necessarily those of the U.S. Census Bureau. The authors thank Ralph Sneider (Monash University) for insightful discussion.

Funding

Support for Li Zhu was provided by the Georgetown Undergraduate Research Opportunities Program (GUROP). Partial funding support for Kimberly Sellers is provided through the ASA/NSF/Census Fellowship Program (U. S. Census Bureau Contract YA1323-14-SE-0122). Galit Shmueli was supported in part by grant 104-2410-H-007-001-MY2 from the Ministry of Science and Technology in Taiwan.