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

Wilasinee Peerajit; Yupaporn Areepong

doi:10.48084/etasr.18603

Authors

Wilasinee Peerajit Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Thailand
Yupaporn Areepong Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Thailand

Volume: 16 | Issue: 3 | Pages: 36840-36848 | June 2026 | https://doi.org/10.48084/etasr.18603

Received: 8 March 2026 | Revised: 18 April 2026 | Accepted: 30 April 2026 | Online: 6 June 2026

Corresponding author: Wilasinee Peerajit

Abstract

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.

Keywords:

Numerical Integral Equation (NIE) method, Average Run Length (ARL), Standard Deviation of the Run Length (SDRL), Median Run Length (MRL), exponential white noise SARFIMA(1,d,1)(1,0,1)12 model

References

D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed. Hoboken, NJ, USA: Wiley, 2019.

S. W. Roberts, "Control Chart Tests Based on Geometric Moving Averages," Technometrics, vol. 1, no. 3, pp. 239–250, Aug. 1959.

E. S. PAGE, "Continuous Inspection Schemes," Biometrika, vol. 41, no. 1–2, pp. 100–115, June 1954.

S. H. Steiner, "EWMA Control Charts with Time-Varying Control Limits and Fast Initial Response," Journal of Quality Technology, vol. 31, no. 1, pp. 75–86, Jan. 1999.

N. Abbas, M. Riaz, and R. J. M. M. Does, "An EWMA-Type Control Chart for Monitoring the Process Mean Using Auxiliary Information," Communications in Statistics - Theory and Methods, vol. 43, no. 16, pp. 3485–3498, Aug. 2014.

M. Awais and A. Haq, "An EWMA chart for monitoring process mean," Journal of Statistical Computation and Simulation, vol. 88, no. 5, pp. 1003–1025, Mar. 2018.

C. W. J. Granger and R. Joyeux, "An Introduction to Long-Memory Time Series Models and Fractional Differencing," Journal of Time Series Analysis, vol. 1, no. 1, pp. 15–29, 1980.

J. R. M. Hosking, "Fractional differencing," Biometrika, vol. 68, no. 1, pp. 165–176, Apr. 1981.

J. Beran, Statistics for Long-Memory Processes. Boca Raton, FL, USA: Chapman and Hall/CRC, 1994.

L. Rabyk and W. Schmid, "EWMA control charts for detecting changes in the mean of a long-memory process," Metrika, vol. 79, no. 3, pp. 267–301, Apr. 2016.

R. T. Baillie, "Long memory processes and fractional integration in econometrics," Journal of Econometrics, vol. 73, no. 1, pp. 5–59, July 1996.

W. Palma, Long‐Memory Time Series: Theory and Methods, 1st ed. Hoboken, NJ, USA: Wiley, 2006.

I. M. S. Pereira and M. Antonia Amaral-Turkman, "Bayesian prediction in threshold autoregressive models with exponential white noise," Test, vol. 13, no. 1, pp. 45–64, June 2004.

P. A. Jacobs and P. a. W. Lewis, "A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1)," Advances in Applied Probability, vol. 9, no. 1, pp. 87–104, Mar. 1977.

Suparman, "A New Estimation Procedure Using a Reversible Jump MCMC Algorithm for AR Models of Exponential White Noise," International Journal of GEOMATE, vol. 15, no. 49, pp. 85–91, Sept. 2018.

C. W. Champ and S. E. Rigdon, "A a comparison of the markov chain and the integral equation approaches for evaluating the run length distribution of quality control charts," Communications in Statistics - Simulation and Computation, vol. 20, no. 1, pp. 191–204, Jan. 1991.

S. V. Crowder, "Average Run Lengths of Exponentially Weighted Moving Average Control Charts," Journal of Quality Technology, vol. 19, no. 3, pp. 161–164, July 1987.

Y. Areepong and W. Peerajit, "Enhancing the Detection Power of CUSUM Charts for Changes in the Mean of the Long-Memory ARFIMAX Model with Exponential White Noise," Engineering, Technology & Applied Science Research, vol. 15, no. 6, pp. 29103–29112, Dec. 2025.

P. Phanthuna and Y. Areepong, "Change-Detected ARIMA (1,1,1) Time Series Using the Approximated and Exact ARL of the MEWMA Scheme," Malaysian Journal of Fundamental and Applied Sciences, vol. 22, no. 1, pp. 215–233, Feb. 2026.

J. S. Hunter, "The Exponentially Weighted Moving Average," Journal of Quality Technology, vol. 18, no. 4, pp. 203–210, Oct. 1986.

D. Brook and D. A. Evans, "An approach to the probability distribution of cusum run length," Biometrika, vol. 59, no. 3, pp. 539–549, Dec. 1972.

C. Li, S. Cui, and D. Wang, "Monitoring the Zero-Inflated Time Series Model of Counts with Random Coefficient," Entropy, vol. 23, no. 3, Mar. 2021, Art. no. 372.

"XAU/USD - Gold Spot US Dollar Historical Data." Investing. https://www.investing.com/currencies/xau-usd-historical-data.

S. Chakraborti, "Run Length Distribution and Percentiles: The Shewhart Chart with Unknown Parameters," Quality Engineering, vol. 19, no. 2, pp. 119–127, Apr. 2007.