Gauss–Legendre Numerical Integrations for Average Run Length Running on EWMA Control Chart with Fractionally Integrated MAX Process

The performance of a process running on an exponentially weighted moving average (EWMA) control chart is contingent upon the ability to detect changes in the process mean rapidly. This entails determining the shortest average run length (ARL) for when a process becomes out-of-control (ARL1). Herein, we propose a numerical integral equation (NIE) method to approximate the ARL for a long-memory fractionally integrated moving-average process with an exogenous variable with underlying exponential white noise running on an EWMA control chart using the Gauss-Legendre quadrature. In a numerical evaluation to compare its performance with that derived by using explicit formulas for this scenario, both performed equally well in terms of accuracy percentage (> 95%) and showed very consistent ARL1 values. Therefore, the NIE approach is acceptable for approximating the ARL for this specific situation. In addition, comparing their standard deviations of the run length (SDRLs) illustrates that the NIE method performed better in rapidly detecting a shift in the process mean. Real data consistent with an FI-MAX process were also analyzed to demonstrate the applicability of using the proposed method for FI-MAX processes on EWMA charts.