Title: On the Statistical Performance of Quality Control Charts with Estimated Parameters

Under estimated in-control parameters, the Phase II control chart performance is expected to vary among practitioners due to the use of different Phase I data sets. Accordingly, the typical measure of Phase II control chart performance, the average run length (ARL), becomes a random variable. In the literature, control charts with estimated parameters were assessed and the appropriate amounts of Phase I data were recommended based on the in-control performance averaged across the practitioner-to-practitioner variability. In this study, aspects of the ARL distribution, such as the standard deviation of the average run length (SDARL) and some quantiles are used to quantify the between-practitioner variability in control charts performance when the process parameters are estimated. It is shown that no realistic amount of Phase I data is sufficient to have confidence that the attained in-control ARL is close to the desired value. Moreover, it is shown that even with the use of larger amounts of historical data, there is still a problem with the excessive false alarm rates. Due to the extreme difficulty of lowering the variation in the in-control ARLs, an alternative design criterion based on the bootstrap approach is recommended for adjusting the control limits. The technique is quite effective in controlling the percentage of short in-control ARLs resulting from the estimation error. Three of the most well-known univariate control charts (Shewhart, EWMA, and CUSUM), and two multivariate charts (T, and MEWMA) are studied.