Modeling and Monitoring of INARCH( <i>p</i> ) Process With Geometrically Inflated Poisson Distribution

ABSTRACT Count data inflated at the first values is common in integer‐valued time series. For example, there will be many values of zero, one and two in the number of infections from a specific disease within a time‐period. Modeling and monitoring this type of data is of paramount importance. In this paper, we propose an ‐geometrically inflated Poisson integer‐valued autoregressive conditional heteroskedasticity (INARCH) model. Some statistical properties are given. Furthermore, we derive estimators using the conditional maximum likelihood (CML) method, while also establishing the asymptotic properties of the CML estimators. In addition, the cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are considered to detect an increasing in the process mean. To verify the effectiveness of the estimation method and the monitoring performance of the two control charts, we perform numerous simulation studies. Finally, two real data examples of crime data and disease data are provided for the practical implementation of the proposed model and control charts.