A stationary negative binomial INAR(1) model based on the beta–binomial thinning operator

Abstract

We propose a stationary negative binomial integer-valued autoregressive model of order one (NB-INAR(1)) built on a beta–binomial thinning operator. Allowing the thinning probability to vary according to a beta distribution induces a flexible survivor mechanism that preserves the binomial mean structure while inflating the conditional variance and capturing latent heterogeneity in persistence. Coupled with suitably specified negative binomial innovations that maintain the marginal distribution, the resulting process forms a Markov chain on the non-negative integers with tractable probabilistic properties, including geometrically decaying autocorrelation and an interpretable lag-one dependence parameter. Parameter estimation is investigated using conditional least squares, method of moments with feasibility diagnostics, and an optional maximum likelihood refinement based on convolution transition probabilities. A Monte Carlo study reveals that innovation parameters become increasingly difficult to identify under extreme persistence and pronounced thinning heterogeneity, underscoring the importance of reporting reliability measures such as feasibility and boundary-clamping rates. An application to weekly road-traffic collision counts (2009–2014) demonstrates the model’s practical performance in fitting and probabilistic forecasting.