Effect of Parameter Estimation on the Performance of Shewhart <img src=image/13425522_01.gif>-joint Chart Looked at in Terms of the Run Length Distribution

Abstract

Using spread-charts to monitor process variation and thereafter using the <img src=image/13425522_02.gif>-chart to monitor the process mean after is a common practice. To apply these charts independently using estimated 3-sigma limits is common. Recently, some authors considered the application of <img src=image/13425522_02.gif> and R-charts together as a charting scheme, <img src=image/13425522_01.gif>-chart when the standards are known, Case KK, only the mean standard is known, Case KU and both standards unknown, Case UU. The average run length (ARL) performance criterion was used. However, because of the skewed nature of the run length (RL) distribution, many authors have frowned at the use of ARL as a sole performance measure and encouraged the percentiles of the RL distribution instead. Therefore, the cdfs of the RLs of the chart under the cases mentioned will be derived in this work, and the percentiles are used to look at the chart for Case KU and the yet to be considered case of the chart, Case UK where only the process variance is known is included for comparison. These are the contribution to the existing literature. <img src=image/13425522_01.gif>-chart performed better in Case KU than in Case UK and the unconditional in-control median run length described the behavior of the chart better than the in-control ARL.