Abstract: The cumulative sum (CUSUM) control chart is widely recognized for its superior ability to detect small to moderate upward shifts in the process mean. This study focuses on accurately approximating the average run length (ARL) of a CUSUM chart applied to a long-memory SARFIMAX(p, d, q, x)x(P, D, Q)s model with exponential white noise. The model accommodates both seasonality and fractional differencing, providing flexibility for complex time-series structures. ARL values are approximated through numerical integral equations (NIEs) using Gaussian quadrature, midpoint, trapezoidal, and Simpson’s rules. Comparative results indicate that all four approaches yield highly consistent ARL estimates, confirming the robustness of the NIE framework. Among them, the midpoint rule demonstrates the lowest computational time while maintaining high accuracy, as reflected by its minimal percentage relative deviation (%Dev). These findings highlight the computational efficiency and sensitivity of the proposed method, supporting its practical applicability in real-world process monitoring scenarios.

Keywords: Average run length (ARL), SAFIMAX(p, d, q, x)×(P, D, Q)s, exponential white noise, numerical integral equation (NIE) method, midpoint rule, trapezoidal rule, Gaussian rule, Simpson's rule

Pages: 231-244