Analytic Solution to ARL Integral Equation of Extended EWMA Control Chart for Seasonal ARX Model

Control charts are important statistical tools for monitoring process stability and identifying any unacceptable changes. The extended exponentially weighted moving average (extended EWMA) control chart, which is an adjustment of the classical EWMA scheme, has been studied for detecting process changes when observations exhibit serial correlation. A basic characteristic in evaluating control chart performance is the average run length (ARL), which can be expressed as a Fredholm integral equation of the second kind and approximated numerically; however, the algorithm of numerical methods is time-consuming. This study aims to propose an analytic solution to the ARL integral equation of the extended EWMA control chart for a seasonal autoregressive model with exogenous variables (SARX) and an exponential white noise. Here, we derived the explicit expression to solve the ARL integral equation and examined its accuracy by numerical integration methods. The primary results showed that the analytic expression provides the accurate ARL values confirmed by the estimated solution and significantly decreases computation time to less than 0.001 seconds. The effectiveness of the extended EWMA control chart was compared to the EWMA and the cumulative sum (CUSUM) procedures. The ARL and other overall performance criteria indicated that the extended EWMA control chart overcomes the conventional control charts in detecting shifts in the process mean under various conditions. Natural gas price data and exchange rate, which are considered exogenous variables, were applied to compare the performance of the control charts.