Integral Equation Solutions to Approximate the Average Run Length of a Long-Memory Seasonal Autoregressive Fractionally Integrated Moving Average Process on an EWMA Control Chart

This study evaluates the performance of the Exponentially Weighted Moving Average (EWMA) control chart for monitoring long-memory seasonal processes using a Numerical Integral Equation (NIE) approach, specifically a Seasonal Autoregressive Fractionally Integrated Moving Average (SARFIMA) process with exponential white noise. The metrics used include the Average Run Length (ARL), Standard Deviation of the Run Length (SDRL), and selected percentiles, which are derived using integral-equation formulations. The NIE method is applied using midpoint, trapezoidal, Simpson's, and Gauss–Legendre quadrature rules, after which monitoring is calibrated to achieve an in-control ARL of approximately 370 for smoothing parameter values of 0.05, 0.10, and 0.15. All quadrature schemes yield identical ARL results, differing only in computational efficiency. As the process mean shift magnitude is increased, the ARL, SDRL, and quartile measures decrease, thereby indicating faster and more stable detection. The practicability of the method is illustrated using monthly XAU/USD gold price data (January 2020 to February 2026), where the fitted SARFIMA model adequately captures seasonal long-memory dynamics and supports reliable EWMA monitoring performance.