A modified Pegram’s operator based autoregressive (mPAR) process for modeling truncated counts

The Pegram’s operator based autoregressive (PAR) model is a well-known method in discrete-valued time series. However, its indicator kernel assigns weight only to repetitions of the previous outcome and does not directly accommodate changes to other outcomes. To address this, we propose a modified PAR (mPAR) process for truncated counts, using a simple kernel that gives positive weight to both staying in the same state and moving to other states. The model is developed for bounded counts with right-truncated Poisson innovations, and we focus mainly on the first-order case. We derive key distributional properties and obtain the h-step predictive distribution and forecasting procedures. Simulation experiments examine maximum likelihood estimation, stability of the estimates, and comparisons with PAR(1) using fit and forecasting criteria. The results show that the estimates stabilize as sample size increases, and that mPAR(1) can offer better fit and improved forecasts than the usual PAR(1) process. Finally, two real datasets are analyzed to illustrate contrasting regimes: one with more frequent changes away from the previous value, where mPAR(1) yields a smaller AIC and some forecasting differences, and another closer to the usual PAR(1) setting, where both models perform similarly.