Average Run Length Computations of Autoregressive and Moving Average Process using the Extended EWMA Procedure

In the past, the control chart served as a statistical tool for detecting process changes. The Exponentially Weighted Moving Average (EWMA) control chart is highly effective for detecting small changes. This research introduces the Extended Exponentially Weighted Moving Average (Extended EWMA) control chart for the Autoregressive and Moving average process with order p = 1 and q = 1 (ARMA(1,1)) The Extended EWMA control chart incorporates two smoothing parameters ( λ1 and λ2 ) derived from the EWMA control chart. A comparative analysis of the performance of the EWMA control chart. The Average Run Length (ARL) value as determined by the explicit formulas in this research, serves as a metric for evaluating the performance of the Extended EWMA control chart and the EWMA control chart. The Numerical Integral Equation (NIE) method is used to verify the accuracy of the ARL for the explicit formulas of the two control charts which has not been before discovered. The effectiveness of control charts can also be evaluated by analyzing SDRL, ARL, MRL, RMI, AEQL, and PCI values as metrics for various design parameter values. After analyzing the results of the ARL and all five performance meters, it was determined that the Extended EWMA control chart is better than the EWMA control chart at all shift sizes of process changes. Finally, the assessment of the ARMA process is being conducted to evaluate the ARL using a dataset on PM2.5 dust levels in Bangkok, Thailand during January and February of 2024.