A first-order spatial integer-valued autoregressive SINAR(1,1) model

Recently, there has been growing interest in modelling non-negative integer-valued time series. Counts of accidents, number of patients admitted to a hospital, number of crimes committed at a particular, counts of transmitted messages and detected errors, etc. are examples of these type of time series. A popular approach in modelling non-negative integer-valued data is by using the binomial thinning operator, which was introduced by Steutal and Van Harn in 1979. The first time series model based on this operator was by Mckenzie (1985). The purpose of this paper is to extend the use of the binomial thinning operator to the spatial case. Specifically, we define the First-Order Spatial Integer-Valued Autoregressive SINAR(1,1) Model and obtain some of its properties namely, the theoretical Autocorrelation Function (ACF), the mean and the variance of this model. In this paper, the Yule-Walker estimates for the parameters of this model are also established. We also simulated some realisations of this process and computed the sample ACF, the sample mean and the sample variance for some selected parameter values. Comparisons were then made with the theoretical ACF, the theoretical mean and variance. Further a simulation study was conducted to evaluate the performance of the proposed Yule-Walker estimates. Finally to illustrate the fitting of the SINAR(1,1) model to a real data set, we considered the yeast cell counts presented by Student (1906). Our results extend the theory and practice of spatial models