The Accurate Results of Average Run Length on Modified EWMA Control Chart for the First-Order Moving Average Process with Exogenous Variables Models

In this study, we derive explicit formulas for the average run length (ARL) of a first-order moving average process with exogenous variables (MAX(1,1)) with exponential white noise and compare their performance on standard and modified and exponentially weighted moving average (EWMA) control charts based on the absolute percentage relative error (APRE) and the relative mean index (RMI). Moreover, we compare the accuracy and CPU time of the explicit formulas with the ARL based on the numerical integral equation (NIE) method derived by using the Gauss-Legendre quadrature rule for the same process on the two control charts. To demonstrate the capability of our explicit formulas approach, we applied it to two real datasets: the closing stock prices for the PTT public company limited with the THB/USD daily foreign exchange rate as the exogenous variable and the monthly gold futures price with the crude oil futures price as the exogenous variable. The results of applying the ARL based on the explicit formulas with the two real datasets show that the modified EWMA control chart performed better than the EWMA control chart under these circumstances.