On the design of Shewhart control charts for count time series under estimation uncertainty

This study is a first step towards a comprehensive analysis of the effects of parameter estimation on the monitoring of autocorrelated count processes. Focusing on Shewhart's c and np charts, various types of count process with different dispersion properties and model structures are considered. The pure effect of discreteness of observations is contrasted with the additional consequences of estimation uncertainty on the control limits of the charts as well as on the resulting run length performance. The main parameter regarding a trustworthy chart design is the actual dispersion of the count process, which, in turn, might be connected to the extent of serial dependence for certain types of process. For some low-dispersion scenarios, the effect of parameter estimation on the design parameters might even fully vanish. The acquired results are illustrated with two real-data examples, where a parametric bootstrap approach is used to judge the reliability of the obtained chart design.