Count Models Based on Weibull Interarrival Times

TLDR

The Poisson and negative binomial models dominate count data analysis because they arise from assuming exponentially distributed interarrival times, yet this assumption limits flexibility and precludes modeling underdispersed counts. This study introduces a computationally tractable Weibull‑based count model that was previously considered infeasible. The model generalizes count data by assuming a Weibull interarrival process, nests the Poisson and negative binomial models as special cases, and achieves closed‑form inference through a polynomial expansion that is conjugate to a gamma heterogeneity distribution. The resulting Weibull count model can represent both over‑ and underdispersed data, incorporate covariates via the hazard function, and be implemented in standard statistical software.

Abstract

The widespread popularity and use of both the Poisson and the negative binomial models for count data arise, in part, from their derivation as the number of arrivals in a given time period assuming exponentially distributed interarrival times (without and with heterogeneity in the underlying base rates, respectively). However, with that clean theory come some limitations including limited flexibility in the assumed underlying arrival rate distribution and the inability to model underdispersed counts (variance less than the mean). Although extant research has addressed some of these issues, there still remain numerous valuable extensions. In this research, we present a model that, due to computational tractability, was previously thought to be infeasible. In particular, we introduce here a generalized model for count data based upon an assumed Weibull interarrival process that nests the Poisson and negative binomial models as special cases. The computational intractability is overcome by deriving the Weibull count model using a polynomial expansion which then allows for closed-form inference (integration term-by-term) when incorporating heterogeneity due to the conjugacy of the expansion and a commonly employed gamma distribution. In addition, we demonstrate that this new Weibull count model can (1) model both over- and underdispersed count data, (2) allow covariates to be introduced in a straightforward manner through the hazard function, and (3) be computed in standard software.