A Reevaluation of the Adaptive Exponentially Weighted Moving Average Control Chart When Parameters are Estimated

Abstract

The performance of control charts can be adversely affected when based on parameter estimates instead of known in‐control parameters. Several studies have shown that a large number of phase I observations may be needed to achieve the desired in‐control statistical performance. However, practitioners use different phase I samples and thus different parameter estimates to construct their control limits. As a consequence, there would be in‐control average run length (ARL) variation between different practitioners. This kind of variation is important to consider when studying the performance of control charts with estimated parameters. Most of the previous literature has relied primarily on the expected value of the ARL (AARL) metric in studying the performance of control charts with estimated parameters. Some recent studies, however, considered the standard deviation of the ARL metric to study the performance of control charts. In this paper, the standard deviation of the ARL metric is used to study the in‐control and out‐of‐control performance of the adaptive exponentially weighted moving average (AEWMA) control chart. The performance of the AEWMA chart is then compared with that of the Shewhart trueX¯ and EWMA control charts. The simulation results show that the AEWMA chart might represent a good solution for practitioners to achieve a reasonable amount of ARL variation from the desired in‐control ARL performance. In addition, we apply a bootstrap‐based design approach that provides protection against frequent false alarms without deteriorating too much the out‐of‐control performance. Copyright © 2014 John Wiley & Sons, Ltd.

The Performance of the Multivariate Adaptive Exponentially Weighted Moving Average Control Chart with Estimated Parameters

“…The bootstrap based‐procedure adjusts the control limits of a control chart to reach an in‐control ARL value which is at least equal to a desired value with a specific probability, say ( α )100 %. According to the previous literature, this procedure is very promising and don't have a severe adverse effect on the out‐of‐control performance of the control chart as shown in several studies (see Aly et al ., and Saleh et al ., for example). Following the same approach of Jones and Steiner and Gandy and Kvaløy's, the practitioner can use the following algorithm to adjust the control limit of the MAEWMA chart: 1Using the available m Phase I samples, each of size n , the practitioner first calculates ; where , for the sub‐grouped MAEWMA chart and , for the individual MAEWMA chart. 2Assuming the true in‐control distribution is N p ( μ 0 , Σ 0 ), generate B bootstrap samples from and calculate the corresponding bootstrap estimates ; i = 1, 2,...., B ; where B is a large number, say 1000. 3Search for the MAEWMA control limit ; i =1,2....., B that satisfies the desired in‐control ARL; where the Phase II data are generated from …”

“…Saleh et al . and Aly et al . give some percentiles for the ARL distribution of some univariate charts to evaluate their performance based on the number of samples used in estimation.…”

“…As mentioned previously, several authors used a bootstrap‐based procedure to overcome the low in‐control ARL values that often occur due to using estimated parameters (see: Gandy and Kvaløy, Aly et al ., Saleh et al ., and Faraz et al …”

“…to evaluate the performance of the EWMA control chart with estimated parameters. Aly et al . compared the performance of the Shewhart , EWMA and the univariate AEWMA charts using this metric.…”

“…designed the EWMA and Shewhart charts using this procedure. Aly et al . used it in adjusting the limits of the univariate AEWMA chart.…”