Treating missing values in INAR(1) models: An application to syndromic surveillance data

Time-series models for count data have found increased interest in recent years. The existing literature refers to the case of data that have been fully observed. In this article, methods for estimating the parameters of the first-order integer-valued autoregressive model in the presence of missing data are proposed. The first method maximizes a conditional likelihood constructed via the observed data based on the k-step-ahead conditional distributions to account for the gaps in the data. The second approach is based on an iterative scheme where missing values are imputed so as to update the estimated parameters. The first method is useful when the predictive distributions have simple forms. We derive in full details this approach when the innovations are assumed to follow a finite mixture of Poisson distributions. The second method is applicable when there are no closed form expression for the conditional likelihood or they are hard to derive. The proposed methods are applied to a dataset concerning syndromic surveillance during the Athens 2004 Olympic Games.