Count Models Based on Weibull Interarrival Times

The widespread popularity and use of both the Poisson and negative binomial models for count data arises, in part, from their derivation as the number of arrivals in a given time period assuming exponenitally distributed interarrival times (without and with heterogeneity in the underlying base rates respectively). However, with that clean theory comes some limitations including limited flexibility in the assumed underlying arrival rate distribution and the inability to model underdispersed counts (variance less than the mean). While 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: (a) sometimes alleviate the need for heterogeneity suggesting that what many think is overdispersion may just be model misfit due to a different and more flexible timing model (Weibull versus exponential), (b) model both over and under dispersed count data, (c) allow covariates to be introduced straightforwardly through the hazard function, and (d) be computed in standard software. In fact, we demonstrate the efficacy of our approach using a data analysis run, including bootstrap standard errors computed via a weighted-likelihood, run in Microsoft Excel