On the Use of the Expected ARL Metric for Control Charts Based on Estimated Parameters

At the origination of the Shewhart control chart, it was assumed that the process parameters were known. The in-control Average Run Length (ARL) and the probability of having a false alarm (P) were introduced as metrics to indicate the in-control performance. These two metrics are related when the process data are i.i.d. normally distributed: the ARL equals 1/P. When the process parameters are unknown and have to be estimated, a similar relation holds for each estimated control chart, but the relation between the expected ARL (the average of the ARLs of all possible estimated charts) and the expectedP is different. Control charts based on estimates are often designed such that the in-control ARL equals a predefined value. This paper shows that the expected in-control ARL is a less suitable design criterion