On the design and performance of upper one-sided Shewhart charts for zero-inflated Poisson processes with estimated parameters

In this work we study the performance of the upper one-sided Shewhart-type control charts for monitoring a zero-inflated Poisson process, in the case of estimated process parameters. In practice, the process parameters are unknown and must be estimated from a Phase I dataset. However, this affects the theoretical performance of a control chart and further investigation is needed to assess how much the performance of the chart is affected. Moreover, practical guidelines are necessary regarding the size of the Phase I sample so as the performance of the control chart in the case of estimated parameters is close to the known parameters case. In this work we use Monte Carlo simulation and evaluate various performance measures that provide useful information on chart's performance in the case of estimated parameters. Both the in-control and the out-of-control performance are investigated. Moreover, we suggest adjustments to obtain the desired in-control performance, when the size of the Phase I sample is prespecified. The results show that very large Phase I samples are needed in order to make the chart's performance, in the case of estimated parameters, close to the theoretical one. Using the suggested adjustments, these differences can be significantly reduced.