Coherent forecasting for over-dispersed time series of count data

Abstract

In the context of an over-dispersed count time series data on disease incidences, we consider the Geometric integer-valued autoregressive process of order 1 or GINAR(1), which was first introduced by McKenzie (Adv. Appl. Probab. 18 (1986) 679–705) as an analogue of continuous AR(1) process with exponential margin (Adv. Appl. Probab. 12 (1980) 727–745) on the positive support ($\mathbb{R}^{+}$). A strong enthusiasm still persists as it is apparent from Ristic et al. (J. Stat. Plann. Inf. 139 (2009) 2218–2226). Coherent forecasting of Poisson INAR(1) process due to Al-Osh and Alzaid (J. Time Ser. Anal. 8 (1987) 261–275) was studied by Freeland and McCabe (Int. J. Forecast. 20 (2004) 427–434). Here, we study the $h$-step ahead forecasting distribution corresponding to GINAR(1) process in details using probability generating function. Large sample distributions of the conditional least squares estimates of the model parameters are derived. Some numerical study is performed to illustrate the theoretical results.