Controlling the exponentially weighted moving average S² control chart false alarm behavior when the in‐control variance level must be estimated

Abstract

Investigating the problem of setting control limits for an exponentially weighted moving average (EWMA) chart 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 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.

New EWMA charts for process variance

“…The adaptive EWMA‐ S 2 charts for dispersion that are effective in detecting a range of shifts in the process dispersion were proposed by Ugaz et al 20 . Knoth 21 studied methods for calibrating the EWMA S 2 chart based on the sample variance S 2 when the in‐control level of the variance must be estimated from the in‐control phase‐I data. Recently, Haq and Razzaq 22 suggested weighted adaptive CUSUM (C) charts for the process variance, called the WAC chart.…”

“…A study on the effect of estimation on several EWMA charts for the variance by considering both the effect of phase-I parameter estimation and robustness to non-normality in the data was made by Zwetsloot and Ajadi 19 . The adaptive EWMA-𝑆 2 charts for dispersion that are effective in detecting a range of shifts in the process dispersion were proposed by Ugaz et al 20 Knoth 21 studied methods for calibrating the EWMA 𝑆 2 chart based on the sample variance 𝑆 2 when the in-control level of the variance must be estimated from the in-control phase-I data. Recently, Haq and Razzaq 22 suggested weighted adaptive CUSUM (C) charts for the process variance, called the WAC chart.…”