Numerical Integral Equation Methods of Average Run Length on EWMA Control Chart for ARMAX(1,1,r) with Exogenous Variables and Application

The control chart is an essential part of statistical quality control (SQC), and it enables real-time monitoring of a wide variety of operations, including hospital outcomes, industrial output, and agricultural yields. This study specifically focuses on using exponentially weighted moving average control charts tailored for autoregressive moving average order (1,1) models with exogenous variables. The performance of these control charts is assessed using the average run length (ARL), estimated through numerical integration. A comparison is made between the ARL values obtained through three methodologies―the midpoint, trapezoidal, and Gaussian rules. Additionally, the CPU time required for the ARL assessment is evaluated. Simulation results indicate that the ARL values achieved using the midpoint and Gaussian rules exhibit similarity, while the trapezoidal approach demonstrates a minor discrepancy of approximately 1℅. Regarding computational efficiency, both the midpoint and trapezoidal approaches outperform Gaussian's rule, with respective time requirements of 4 - 6 seconds and 40 - 43 seconds. Furthermore, the proposed methodology is applied to real-world data about crude oil prices.