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

Abstract

SUMMARYThis 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 ≥ 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.

Challenges in designing a disease surveillance plan: What we have and what we need?

“…These models are adequate for providing one-step-ahead forecasts but are not adequate for longer range forecasting. Sound alternatives in these situations are provided by Pavlopoulos and Karlis (2007), Leonenko et al (2007), Weiß (2009aWeiß ( , 2009b, Wu (2009), andZhu (2011). Integer valued autoregressive model INAR(p) and the more general integer-valued GARCH models: These models are an integer (counts) version of the continuous-valued autoregressive AR(p) times series models (see, e.g., AlOsh and Alzaid, 1986;Jin-Guan and Yuan, 1991).…”

Treating Missing Values in INAR(1) Models

“…For certain models we derive the conditional distribution based on the results of Freeland and McCabe (2004). For this approach we consider a general model by assuming that the innovations follow a finite mixture of Poisson distribution (see Pavlopoulos and Karlis (2007)) and we fully derive the conditional distributions needed.…”

“…To disentangle the problem we will consider finite Poisson mixtures innovations as in Pavlopoulos and Karlis (2007) and we will show that in this case the predictive distributions belong to a specific family of distributions that allow for relatively easy computations (see Section 3).…”

Asymptotic behavior of unstable INAR(p) processes

“…The broad scope of the empirical literature in which INAR models are applied indicates its relevance. Examples of such applications include [16] (epileptic seizure counts), [5] (longitudinal count data), [41] (rainfall measurements), [7,38] (economics), [8] (finance), [19] (insurance), [35] (environmental studies) and [32] (finance and mortality).…”