Are estimated control charts in control?

Abstract

Standard control chart practice typically assumes normality and uses estimated parameters using data from an in-control process. However, because of the extreme quantiles involved, large relative errors will result for common performance characteristics such as the out-of-control signal probability or the average run length. Due to the estimation, such performance characteristics are stochastic and hence the relative errors involved can be analyzed in various ways. To assess the effects of these various ways of estimation, we look at some exceedance probabilities. It is demonstrated how corrections can be derived to bring the estimated false alarm rates close to their nominal values. Exact results are given, followed by simple approximations. The latter reveal the way in which the corrections depend on the underlying parameters, thus allowing a sensible approach in practice. Some illustration is provided, as well as a brief analysis of the out-of-control behavior of the corrected charts.

Keywords:

• Statistical process control
• Shewhart
• Exceedance probability
• Tolerance region

Acknowledgements

The authors appreciated the comments by the referees, which resulted in a substantially improved presentation. This research was supported by the Technology Foundation STW, applied science division of NWO and the technology programme of the Ministry of Economic Affairs.