Development of a Numerical Approach to Evaluate the Performance of the CUSUM Control Chart for a Long Memory SARFIMA Model

Assessing the performance of the cumulative sum (CUSUM) control chart is a vital component of statistical process control. This assessment frequently necessitates the approximation of the average run length (ARL) through numerical integration equation (NIE) approaches involving the Midpoint, Trapezoidal, Gauss-Legendre Quadrature Rule, or Simpson's Rule. The focus of this study is evaluating the performance of the CUSUM control chart for a seasonal time series process, specifically long memory Seasonal Autoregressive Fractionally Integrated Moving Average (long memory SARFIMA(1, 0, 1)(1, D, 1)L) model with exponential white noise. These approaches can be used to approximate the ARL by solving the Fredholm integral equation of the second kind. The performance evaluation in terms of the out-of-control ARL (ARL1) and computational time indicates that all four approaches diminished swiftly for minor shifts in the process mean and exhibited no significant differences across various SARFIMA(1, 0, 1)(1, D, 1)L models. The Midpoint Rule demonstrated the fastest computation, closely followed by that using the Trapezoidal Rule, while utilizing the Gauss-Legendre Quadrature Rule or Simpson's Rule made the computation considerably slower. In summary, the Midpoint Rule provides a sufficiently accurate and computationally efficient approach for assessing the CUSUM control chart with a seasonal time series process and is appropriate for practical data applications.