Derivation of Analytical Solution to ARL Integral Equation of the Extended EWMA Control Chart for Time Series ARMAX Process

Control charts are effective tools in statistical process control that are used for monitoring process quality and detecting unusual changes. The extended exponentially weighted moving average (extended EWMA) control chart is one of a traditional EWMA-type improvement which showed a greater performance in detecting changes in autocorrelated process. Average run length (ARL) is a basic performance metric for measuring the efficiency of statistical control charts. ARL can be expressed as a Fredholm integral equation of the second kind and approximated numerically. This study aims to propose an analytical solution to the ARL integral equation of the extended EWMA control chart for an autoregressive moving average model with exogenous variables (ARMAX) and an exponential white noise. The analytical solution of the ARL integral equation was derived and compared with the accuracy to numerical integration method. The simulation results indicated that the analytical expression provides accurate ARL values and significantly decreases computation time to less than 0.001 seconds. The comparisons on the ARL and other overall performance criteria for the extended EWMA, the EWMA, and the cumulative sum (CUSUM) procedures illustrated that the extended EWMA control chart outperforms the others in detecting the process mean changes.