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

To analyse count time series data inflated at the <i>r</i> + 1 values {0,1,…,r}, we propose a new first-order integer-valued autoregressive process with <i>r</i>-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.

针对在r+1个点{0,1,…,r}处存在膨胀的计数时间序列数据，本文提出一种带有r几何膨胀泊松新息项的一阶整值自回归过程。文中给出了该过程的若干统计性质及条件极大似然估计方法。为开展统计监控研究，本文针对所提出的过程重点探讨了累积和控制图、指数加权移动平均控制图以及联合跳跃控制图。数值模拟结果表明，该条件极大似然估计量具有无偏性。此外，在实际应用中，累积和控制图是监控该模型的最优选择。文末通过电话投诉数据的实际应用案例，对所提出的方法进行了实例说明。