A non-stationary integer-valued autoregressive model

It is frequent to encounter a time series of counts which are small in value and show a trend having relatively large fluctuation. To handle such a non-stationary integer-valued time series with a large dispersion, we introduce a new process called integer-valued autoregressive process of order p with signed binomial thinning (INARS(p)). This INARS(p) uniquely exists and is stationary under the same stationary condition as in the AR(p) process. We provide the properties of the INARS(p) as well as the asymptotic normality of the estimates of the model parameters. This new process includes previous integer-valued autoregressive processes as special cases. To preserve integer-valued nature of the INARS(p) and to avoid difficulty in deriving the distributional properties of the forecasts, we propose a bootstrap approach for deriving forecasts and confidence intervals. We apply the INARS(p) to the frequency of new patients diagnosed with acquired immunodeficiency syndrome (AIDS) in Baltimore, Maryland, U.S. during the period of 108 months from January 1993 to December 2001.