Modelling and monitoring of INAR(1) process with geometrically inflated Poisson innovations

To analyse count time series data inflated at the r + 1 values $ \{0,1,\ldots ,r\} $ {0,1,…,r}, we propose a new first-order integer-valued autoregressive process with r-geometrically inflated Poisson innovations. Some statistical properties together with conditional maximum likelihood estimate are provided. For the purpose of statistical monitoring, we focus on the cumulative sum chart, exponentially weighted moving average chart and combined jumps chart towards the proposed process. Numerical simulations indicate that the conditional maximum likelihood estimator is unbiased. Moreover, the cumulative sum chart is the best choice to monitor our model in practice. Some applications about telephone complaints data are provided to illustrate the proposed methods.