Abstract
Accurate calculation of the Average Run Length (ARL) for exponentially weighted moving average (EWMA) charts might be a tedious task. The omnipresent Markov chain approach is a common and effective tool to perform these calculations — see Lucas and Saccucci (1990) and Saccucci and Lucas (1990) for its application in case of EWMA charts. However, Crowder (1987b) and Knoth (2005) provided more sophisticated methods from the rich literature of numerical analysis to solve the ARL integral equation. These algorithms lead to very fast implementations for determining the ARL with high accuracy such as Crowder (1987a), or the R package spc (Knoth 2019) with its functions xewma.arl() and sewma.arl(). Crowder (1987a) utilized the popular Nyström method (Nyström 1930) which fails for bounded random variables existing, for example, in the case of an EWMA chart monitoring the variance. For the latter, Knoth (2005) utilized the so-called collocation method. It turns out that the numerical problems are even more severe for beta distributed random variables, which are bounded from both sides, typically on (0, 1). We illustrate these subtleties and provide extensions from Knoth (2005) to achieve high accuracy in an efficient way.
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References
Brezinski, C. (1985). Convergence acceleration methods: The past decade. Journal of Computational and Applied Mathematics, 12–13, 19–36. https://doi.org/10.1016/0377-0427(85)90005-6.
Brook, D., & Evans, D. A. (1972). An approach to the probability distribution of CUSUM run length. Biometrika, 59(3), 539–549. https://doi.org/10.2307/2334805.
Castagliola, P. (2001). An (\(\tilde{X}/R\))-EWMA control chart for monitoring the process sample median. International Journal of Reliability, Quality, and Safety Engineering, 8(2), 123–135. https://doi.org/10.1142/S0218539301000414.
Crowder, S. V. (1987a). Average run lengths of exponentially weighted moving average control charts. Journal of Quality Technology, 19(3), 161–164. https://doi.org/10.1080/00224065.1987.11979055.
Crowder, S. V. (1987b). A simple method for studying run-length distributions of exponentially weighted moving average charts. Technometrics, 29(4), 401–407. https://doi.org/10.1080/00401706.1987.10488267.
Crowder, S. V., & Hamilton, M. D. (1992). An EWMA for monitoring a process standard deviation. Journal of Quality Technology, 24(1), 12–21. https://doi.org/10.1080/00224065.1992.11979369.
Forbes, C., Evans, M., Hasting, N., & Peacock, B. (2011). Statistical Distributions (4th ed.). New York: Wiley.
Gan, F. F. (1993). Exponentially weighted moving average control charts with reflecting boundaries. Journal of Statistical Computation and Simulation, 46(1–2), 45–67. https://doi.org/10.1080/00949659308811492.
Gan, F. F., & Tan, T. (2010). Risk-adjusted number-between failures charting procedures for monitoring a patient care process for acute myocardial infarctions. Health Care Management Science, 13(3), 222–233. https://doi.org/10.1007/s10729-010-9125-8.
Gan, F. F., Lin, L., & Loke, C. K. (2012). Risk-adjusted cumulative sum charting procedures. In H. J. Lenz, W. Schmid, & P. T. Wilrich (Eds.), Frontiers in Statistical Quality Control 10 (pp. 207–225). Heidelberg: Physica (Springer). https://doi.org/10.1007/978-3-7908-2846-7.
Gianino, A. B., Champ, C. W., & Rigdon, S. E. (1990). Solving integral equations by the collocation method. In ASA Proceedings of the Statistical Computing Section, American Statistical Association (pp. 101–102)
Graham, M. A., Mukherjee, A., & Chakraborti, S. (2012). Distribution-free exponentially weighted moving average control charts for monitoring unknown location. Computational Statistics and Data Analysis, 56(8), 2539–2561. https://doi.org/10.1016/j.csda.2012.02.010.
Grigg, O., & Spiegelhalter, D. (2007). A simple risk-adjusted exponentially weighted moving average. Journal of the American Statistical Association, 102(477), 140–152. https://doi.org/10.1198/016214506000001121.
Hawkins, D. M. (1992). Evaluation of average run lengths of cumulative sum charts for an arbitrary data distribution. Communications in Statistics - Simulation and Computation, 21(4), 1001–1020. https://doi.org/10.1080/03610919208813063.
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions (2nd ed., Vol. 2)., Wiley Series in Probability and Statistics New York: Wiley.
Knoth, S. (2005). Accurate ARL computation for EWMA-\(S^2\) control charts. Statistics and Computing, 15(4), 341–352. https://doi.org/10.1007/s11222-005-3393-z.
Knoth, S. (2006). Computation of the ARL for CUSUM-\(S^2\) schemes. Computational Statistics and Data Analysis, 51(2), 499–512. https://doi.org/10.1016/j.csda.2005.09.015.
Knoth, S. (2019). spc: Statistical Process Control - Collection of Some Useful Functions. R package version 0.6.1.
Lahcene, B. (2013). On Pearson families of distributions and its applications. African Journal of Mathematics and Computer Science Research, 6(5), 108–117. https://doi.org/10.5897/AJMCSR2013.0465.
Loke, C. K., & Gan, F. F. (2012). Joint monitoring scheme for clinical failures and predisposed risks. Quality Technology and Quantitative Management, 9(1), 3–21. https://doi.org/10.1080/16843703.2012.11673274.
Lucas, J. M. (1982). Combined Shewhart-CUSUM quality control schemes. Journal of Quality Technology, 14(2), 51–59. https://doi.org/10.1080/00224065.1982.11978790.
Lucas, J. M., & Saccucci, M. S. (1990). Exponentially weighted moving average control schemes: Properties and enhancements. Technometrics, 32(1), 1–12. https://doi.org/10.1080/00401706.1990.10484583.
Nyström, E. J. (1930). Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben. Acta Mathematica, 54(1), 185–204. https://doi.org/10.1007/BF02547521.
Page, E. S. (1954). Control charts for the mean of a normal population. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 16(1), 131–135. https://doi.org/10.1111/j.2517-6161.1954.tb00154.x.
Pearson, K. (1895). X. Contributions to the mathematical theory of evolution.–II. Skew variation in homogeneous material. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 186, 343–414. https://doi.org/10.1098/rsta.1895.0010.
Pearson, K. (1916). IX. Mathematical contributions to the theory of evolution.–XIX. Second supplement to a memoir on skew variation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 216, 429–457. https://doi.org/10.1098/rsta.1916.0009.
Reynolds, M. R., & Stoumbos, Z. G. (2005). Should exponentially weighted moving average and cumulative sum charts be used with Shewhart limits? Technometrics, 47(4), 409–424. https://doi.org/10.1198/004017005000000382.
Roberts, S. W. (1959). Control chart tests based on geometric moving averages. Technometrics, 1(3), 239–250. https://doi.org/10.1080/00401706.1959.10489860.
Saccucci, M. S., & Lucas, J. M. (1990). Average run lengths for exponentially weighted moving average control schemes using the Markov chain approach. Journal of Quality Technology, 22(2), 154–162. https://doi.org/10.1080/00224065.1990.11979227.
Waldmann, K. H. (1986). Bounds for the distribution of the run length of geometric moving average charts. Journal of the Royal Statistical Society: Series C (Applied Statistics), 35(2), 151–158. https://doi.org/10.2307/2347265.
Woodall, W. H., & Mahmoud, M. A. (2005). The inertial properties of quality control charts. Technometrics, 47(4), 425–436. https://doi.org/10.1198/004017005000000256.
Yashchin, E. (1987). Some aspects of the theory of statistical control schemes. IBM Journal of Research and Development, 31(2), 199–205. https://doi.org/10.1147/rd.312.0199.
Yashchin, E. (1989). Weighted cumulative sum technique. Technometrics, 31(1), 321–338. https://doi.org/10.1080/00401706.1989.10488555.
Yashchin, E. (2019). Gradient analysis of Markov-type control schemes and its applications. Communications in Statistics – Simulation and Computation, online: 1–23. https://doi.org/10.1080/03610918.2019.1687718
Yousry, M. A., Sturm, G. W., Feltz, C. J., & Noorossana, R. (1991). Process monitoring in real time: Empirical Bayes approach - discrete case. Quality and Reliability Engineering International, 7(3), 123–132. https://doi.org/10.1002/qre.4680070303.
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Knoth, S. (2021). On the Calculation of the ARL for Beta EWMA Control Charts. In: Knoth, S., Schmid, W. (eds) Frontiers in Statistical Quality Control 13. ISQC 2019. Frontiers in Statistical Quality Control. Springer, Cham. https://doi.org/10.1007/978-3-030-67856-2_3
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