INAR(1) modeling of overdispersed count series with an environmental application

Abstract

AbstractThis paper is concerned with a novel version of the INAR(1) model, a non‐linear auto‐regressive Markov chain on ℕ, with innovations following a finite mixture distribution of $m \geq 1$ Poisson laws. For $m > 1$, the stationary marginal probability distribution of the chain is overdispersed relative to a Poisson, thus making INAR(1) suitable for modeling time series of counts with arbitrary overdispersion. The one‐step transition probability function of the chain is also a finite mixture, of m Poisson‐Binomial laws, facilitating likelihood‐based inference for model parameters. An explicit EM‐algorithm is devised for inference by maximization of a conditional likelihood. Alternative options for inference are discussed along with criteria for selecting m. Integer‐valued prediction (IP) is developed by a parametric bootstrap approach to ‘coherent’ forecasting, and a certain test statistic based on predictions is introduced for assessing performance of the fitted model. The proposed model is fitted to time series of counts of pixels where spatially averaged rain rate exceeds a given threshold level, illustrating its capabilities in challenging cases of highly overdispersed count data. Copyright © 2007 John Wiley & Sons, Ltd.