Multivariate time between events control charts for Gumbel's bivariate exponential distribution with estimated parameters

The traditional approach of statistical process control is to estimate the in-control parameters from reference samples obtained in Phase I and then apply them to Phase II monitoring, which affects the performance of the monitoring scheme. We analyzed the effect of parameter estimation on three multivariate time between events control charts for monitoring the Gumbel's bivariate exponential distribution, namely the joint multivariate exponentially weighted moving average and multivariate rate, the joint multivariate cumulative sum and multivariate rate, and the joint paired individual exponentially weighted moving average and multivariate rate control charts. The simulation results show that the performance of the control charts is severely affected if control limits with known parameters are used, especially when only a few historical sample data are available. Mainly, the steady-state average time to signal was used as an evaluation index to analyze the affected degree of the three control charts, and the control limits were corrected. In addition, we evaluate the effect of model misspecification on the control charts. Finally, an example is given to illustrate the application of the proposed method.