Run-length performance of AE-I and AE-II mean charts under phase-I parameter estimation

Abstract

Adaptive exponentially weighted moving average charts are effective tools for detecting small-to-moderate shifts in a process mean. In practical applications, however, the incontrol mean and standard deviation are usually unknown and must be estimated from Phase I data before Phase II monitoring begins. This estimation uncertainty can affect both false-alarm behaviour and out-of-control detection performance. Although the adaptive exponentially weighted moving average-I chart has been studied under parameter estimation, the corresponding run-length behaviour of the parameter-free adaptive exponentially weighted moving average-II chart has not been systematically examined. This study evaluates the adaptive exponentially weighted moving average-II mean chart when process parameters are estimated from Phase I samples and compares it with the adaptive exponentially weighted moving average-I chart under a common in-control average run length calibration. Monte Carlo simulations with 100,000 replications are conducted to obtain zero-state and steady-state run-length profiles for different Phase I sample sizes and mean-shift magnitudes. Overall performance is summarized using integrated relative average run length and relative mean index measures. In addition to the normal model, a heavy-tailed t-distribution setting is considered as a robustness check. We also use a real industrial piston-ring dataset to illustrate practical implementation. The results show that parameter-estimation effects are strongest when the number of Phase I subgroups is small and diminish as m increases. Under calibrated in-control performance, the adaptive exponentially weighted moving average-II chart generally detects small-to-moderate shifts faster, whereas the adaptive exponentially weighted moving average-I chart remains competitive for moderate and mixed-range shifts.

Keywords

• Adaptive EWMA
• average run length
• parameter estimation
• process mean monitoring
• simulation
• statistical process control.

References

1. [1] W. Albers and W.C.M. Kallenberg, Are estimated control charts in control?, Statistics 38 (1), 67–79, 2004.
2. [2] H. AlQadi, W. Abdelfattah, T. Abbas, A.A.H. Ahmadini, A.I.A. Sayed and B. Ahmad, Design of an EWMA control chart by adaptation of smoothing constant based on a function of estimated shift, Sci. Rep. 15 (1), 24730, 2025.
3. [3] A.A. Aly, N.A. Saleh, M.A. Mahmoud and W.H. Woodall, A reevaluation of the adaptive EWMA control chart when parameters are estimated, Qual. Reliab. Eng. Int. 31 (8):1611–1622, 2015.
4. [4] G. Capizzi and G. Masarotto, An adaptive exponentially weighted moving average control chart, Technometrics 45 (3), 199–207, 2003.
5. [5] G. Chen, The mean and standard deviation of the run length distribution of $\bar{X}$ charts when control limits are estimated, Stat. Sin. 7 (3), 789–798, 1997.
6. [6] Y. Dai, Y. Luo, Z. Li and Z.Wang, A new adaptive CUSUM control chart for detecting the multivariate process mean, Qual. Reliab. Eng. Int. 27 (7), 877–884, 2011.
7. [7] N. M. Gunaime, Adaptive EWMA control chart using MLP-ANN for real-time shift detection, Qual. Reliab. Eng. Int. 41 (7), 3137–3160, 2025.
8. [8] D. Han and F. Tsung, A reference-free CUSCORE chart for dynamic mean change detection and a unified framework for charting performance comparison, J. Am. Stat. Assoc. 101 (473), 368–386, 2006.

9. [9] A. Haq, A new adaptive EWMA control chart using auxiliary information for monitoring the process mean, Commun. Stat. Theory Methods 47 (19), 4840–4858, 2018.
10. [10] A. Haq, R. Gulzar and M. B. C. Khoo, An efficient adaptive EWMA control chart for monitoring the process mean, Qual. Reliab. Eng. Int. 34 (4), 563–571, 2018.
11. [11] A. Haq and M. B. C. Khoo, A parameter-free adaptive EWMA mean chart, Qual. Technol. Quant. Manag. 17 (5), 528–543, 2020.
12. [12] W. A. Jensen, L. A. Jones-Farmer, C. W. Champ and W. H. Woodall, Effects of parameter estimation on control chart properties: a literature review, J. Qual. Technol. 38 (4), 349–364, 2006.
13. [13] W. Jiang, L. Shu and D. W. Apley, Adaptive CUSUM procedures with EWMA-based shift estimators, IIE Trans. 40 (10), 992–1003, 2008. ,
14. [14] R. Jiang, J. Zhang and Z. Yu, Adaptive EWMA control chart for monitoring twoparameter exponential distribution with Type-II right censored data, J. Stat. Comput. Simul. 94 (4), 787–819, 2024.
15. [15] L.A. Jones, The statistical design of EWMA control charts with estimated parameters, J. Qual. Technol. 34 (3), 277–288, 2002.
16. [16] L.A. Jones, C.W. Champ and S.E. Rigdon, The run length distribution of the CUSUM chart with estimated parameters, J. Qual. Technol. 36 (1), 95–108, 2004.
17. [17] L.A. Jones, C.W. Champ and S.E. Rigdon, The performance of exponentially weighted moving average charts with estimated parameters, Technometrics 43 (2), 156–167, 2001.
18. [18] I. Khan, M. Noor-ul-Amin, D.M. Khan, S.A. AlQahtani and W. Sumelka, Adaptive EWMA control chart using Bayesian approach under ranked set sampling schemes with application to Hard Bake process, Sci. Rep. 13 (1), 9463, 2023.
19. [19] S.-L. Lu, M.-C. Chen, J.-H. Chen, S.-F. Yang and K. Chatterjee, Considering AEWMA control chart applied to Gamma-distributed data with fixed and variable sampling intervals, Sci. Rep. 16 (1),1684, 2026.
20. [20] J.M. Lucas and M.S. Saccucci, Exponentially weighted moving average control schemes: properties and enhancements, Technometrics 32 (1), 1–12, 1990.
21. [21] D.C. Montgomery, Introduction to Statistical Quality Control, 6th ed., John Wiley & Sons, Hoboken, NJ, 2009.
22. [22] I.A. Nafisah, M.M.A. Almazah, A.Y. Al-Rezami and S. Hussain, Variable sample size based EWMA control chart with an exponential scaling mechanism for production process monitoring, Sci. Rep. 15 (1), 30964, 2025.
23. [23] M. Noor-ul-Amin, M.W. Kazmi, S. Alkhalaf, S. Abdel-Khalek and M. Nabi, Machine learning based parameter-free adaptive EWMA control chart to monitor process dispersion, Sci. Rep. 14 (1), 31271, 2024.
24. [24] K.H. Ong, S.Y. Teh, K.L. Soh, Z.L. Chong and M.B.C. Khoo, Optimal design of S2-EWMA control chart with estimated parameters based on median run length and expected median run length, Qual. Reliab. Eng. Int. 41 (4), 1495–1532, 2025.
25. [25] E.S. Page, Continuous inspection schemes, Biometrika 41 (1/2), 100–115, 1954.
26. [26] S. Psarakis, A.K. Vyniou and P. Castagliola, Some recent developments on the effects of parameter estimation on control charts, Qual. Reliab. Eng. Int. 30 (8), 1113–1129, 2014.
27. [27] A. Riaz, M. Noor-ul-Amin, W. Emam, Y. Tashkandy, U. Yasmeen and J. Rahimi, Adaptive EWMA control chart for monitoring the coefficient of variation under ranked set sampling schemes, Sci. Rep. 13 (1), 17617, 2023.
28. [28] S.W. Roberts, Control chart tests based on geometric moving averages, Technometrics 1 (3), 239–250, 1959.
29. [29] N.A. Saleh, M.A. Mahmoud and A.S.G. Abdel-Salam, The performance of the adaptive exponentially weighted moving average control chart with estimated parameters, Qual. Reliab. Eng. Int. 29 (4), 595–606, 2013.
30. [30] N.A. Saleh, M.A. Mahmoud, L.A. Jones-Farmer, I. Zwetsloot and W.H. Woodall, Another look at the EWMA control chart with estimated parameters, J. Qual. Technol. 47 (4), 363–382, 2015.
31. [31] L. Shu, W. Jiang and Z. Wu, Adaptive CUSUM procedures with Markovian mean estimation, Comput. Stat. Data Anal. 52 (9), 4395–4409, 2008.
32. [32] B.F.T. Simoes, E.K. Epprecht and A.F.B. Costa, Performance comparisons of EWMA control chart schemes, Qual. Technol. Quant. Manag. 7 (3), 249–261, 2010.
33. [33] R.S. Sparks, CUSUM charts for signalling varying location shifts, J. Qual. Technol. 32 (2), 157–171, 2000.
34. [34] A. Tang, P. Castagliola, J. Sun and X. Hu, The effect of measurement errors on the adaptive EWMA chart, Qual. Reliab. Eng. Int. 34 (4), 609–630, 2018.
35. [35] K.P. Tran, P. Castagliola and N. Balakrishnan, On the performance of the Shewhart median chart in the presence of measurement errors, Qual. Reliab. Eng. Int. 33 (5), 1019–1029, 2017.
36. [36] F. Tsung and K. Wang, Adaptive charting techniques: literature review and extensions, in: Frontiers in Statistical Quality Control 9, H.-J. Lenz, P.-T. Wilrich and W. Schmid (eds.), 19–35, Physica-Verlag, Heidelberg, 2010.
37. [37] W.H. Woodall and M.A. Mahmoud, The inertial properties of quality control charts, Technometrics 47 (4), 425–436, 2005.
38. [38] Z. Wu, J. Jiao, M. Yang, Y. Liu and Z. Wang, An enhanced adaptive CUSUM control chart, IIE Trans. 41 (7), 642–653, 2009.
39. [39] A.A. Zaagan, H.M. Aljohani, H. Zogan and M. Nabi, A machine learning assisted truncated adaptive EWMA chart for robust process monitoring, Sci. Rep. 16 (1), 3196, 2026.
40. [40] Y. Zhao, F. Tsung and Z. Wang, Dual CUSUM control schemes for detecting a range of mean shifts, IIE Trans. 37 (11), 1047–1057, 2005.