Controlling the EWMA $S^2$ control chart false alarm behavior when the in-control variance level must be estimated

TL;DR

This paper develops numerical algorithms to adjust EWMA S^2 control chart limits for better false alarm control when the in-control variance must be estimated, considering parameter uncertainty and using a false alarm probability approach.

Contribution

It introduces a novel method for adjusting control limits based on false alarm probability, with algorithms implemented in an R package for variance monitoring.

Findings

Adjusted control limits improve false alarm performance.

The approach differs significantly from unconditional ARL control.

Impact of smoothing constant and sample size on chart performance is analyzed.

Abstract

Investigating the problem of setting control limits in the case of parameter uncertainty is more accessible when monitoring the variance because only one parameter has to be estimated. Simply ignoring the induced uncertainty frequently leads to control charts with poor false alarm performances. Adjusting the unconditional in-control (IC) average run length (ARL) makes the situation even worse. Guaranteeing a minimum conditional IC ARL with some given probability is another very popular approach to solving these difficulties. However, it is very conservative as well as more complex and more difficult to communicate. We utilize the probability of a false alarm within the planned number of points to be plotted on the control chart. It turns out that adjusting this probability produces notably different limit adjustments compared to controlling the unconditional IC ARL. We then develop numerical algorithms to determine the respective modifications of the upper and two-sided exponentially weighted moving average (EWMA) charts based on the sample variance for normally distributed data. These algorithms are made available within an R package. Finally, the impacts of the EWMA smoothing constant and the size of the preliminary sample on the control chart design and its performance are studied.