Nonparametric control limits incorporating exceedance probability criterion for statistical process monitoring with commonly employed small to moderate sample sizes

References

• AIAG. 2005. Statistical Process Control (SPC)-reference manual. 2nd ed. Detroit, MI: Automotive Industry Action Group.
   Google Scholar
• Albers, W., and W. C. M. Kallenberg. 2004a. Are estimated control charts in control? Statistics 38 (1):67–79. doi:10.1080/02669760310001619369.
   Web of Science ®Google Scholar
• Albers, W., and W. C. M. Kallenberg. 2004b. Estimation in Shewhart control charts: Effects and corrections. Metrika 59 (3):207–34. doi:10.1007/s001840300280.
   Web of Science ®Google Scholar
• Becker, M., S. Klößner, and M. M. Becker. 2017. Package ‘PearsonDS. Australian & New Zealand 50 (2):199–205.
   Google Scholar
• Brown, L. D., T. T. Cai, and A. DasGupta. 2001. Interval estimation for a binomial proportion. Statistical Science 16 (2):101–33. doi:10.1214/ss/1009213286.
   Web of Science ®Google Scholar
• Cai, T. T., and H. Wang. 2009. Tolerance intervals for discrete distributions in exponential families. Statistica Sinica 19 (3):905–23.
   Web of Science ®Google Scholar
• Capizzi, G., and G. Masarotto. 2020. Guaranteed in-control control chart performance with cautious parameter learning. Journal of Quality Technology 52 (4):385–403. doi:10.1080/00224065.2019.1640096.
   Web of Science ®Google Scholar
• Chakraborti, S., S. W. Human, and M. A. Graham. 2008. Phase I statistical process control charts: An overview and some results. Quality Engineering 21 (1):52–62. doi:10.1080/08982110802445561.
   Web of Science ®Google Scholar
• Chen, G. 1997. The mean and standard deviation of the run length distribution of X charts when control limits are estimated. Statistica Sinica 7 (3):789–98.
   Web of Science ®Google Scholar
• Diebold, A. C. 2001. Handbook of silicon semiconductor metrology. USA: CRC Press.
   Google Scholar
• Diko, M. D., S. Chakraborti, and R. J. Does. 2019. Guaranteed in-control performance of the EWMA chart for monitoring the mean. Quality and Reliability Engineering International 35 (4):1144–60. doi:10.1002/qre.2450.
   Web of Science ®Google Scholar
• Epprecht, E. K., L. D. Loureiro, and S. Chakraborti. 2015. Effect of the amount of phase I data on the phase II performance of S 2 and S control charts. Journal of Quality Technology 47 (2):139–55. doi:10.1080/00224065.2015.11918121.
   Web of Science ®Google Scholar
• Goedhart, R., M. Schoonhoven, and R. J. Does. 2017. Guaranteed in-control performance for the Shewhart X and X control charts. Journal of Quality Technology 49 (2):155–71. doi:10.1080/00224065.2017.11917986.
   Web of Science ®Google Scholar
• Goedhart, R., M. Schoonhoven, and R. J. Does. 2020. Nonparametric control of the conditional performance in statistical process monitoring. Journal of Quality Technology 52 (4):355–69. doi:10.1080/00224065.2019.1611352.
   Web of Science ®Google Scholar
• Goedhart, R., M. Schoonhoven, I. M. Zwetsloot, and R. J. Does. 2017. Discussion of “Bridging the gap between theory and practice in basic statistical process monitoring. Quality Engineering 29 (1):18–21.
   Web of Science ®Google Scholar
• Goldman, M., and D. M. Kaplan. 2018. Non-parametric inference on (conditional) quantile differences and interquantile ranges, using L-statistics. The Econometrics Journal 21 (2):136–69. doi:10.1111/ectj.12095.
   Web of Science ®Google Scholar
• Hutson, A. D. 2002. A semi-parametric quantile function estimator for use in bootstrap estimation procedures. Statistics and Computing 12 (4):331–8. doi:10.1023/A:1020783911574.
   Web of Science ®Google Scholar
• Jardim, F. S., S. Chakraborti, and E. K. Epprecht. 2019. Chart with estimated parameters: The conditional ARL distribution and new insights. Production and Operations Management 28 (6):1545–57. doi:10.1111/poms.12985.
   Web of Science ®Google Scholar
• Jardim, F. S., S. Chakraborti, and E. K. Epprecht. 2020. Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart X Chart. Journal of Quality Technology 52 (2):198–217. doi:10.1080/00224065.2019.1571345.
   Web of Science ®Google Scholar
• Jensen, W. A., L. A. Jones-Farmer, C. W. Champ, and W. H. Woodall. 2006. Effects of parameter estimation on control chart properties: A literature review. Journal of Quality Technology 38 (4):349–64. doi:10.1080/00224065.2006.11918623.
   Web of Science ®Google Scholar
• Jones, M. C. 2002. On fractional uniform order statistics. Statistics & Probability Letters 58 (1):93–6. doi:10.1016/S0167-7152(02)00119-0.
   Web of Science ®Google Scholar
• Jones-Farmer, L. A., W. H. Woodall, S. H. Steiner, and C. W. Champ. 2014. An overview of phase I analysis for process improvement and monitoring. Journal of Quality Technology 46 (3):265–80. doi:10.1080/00224065.2014.11917969.
   Web of Science ®Google Scholar
• Krishnamoorthy, K., and T. Mathew. 2009. Statistical tolerance regions: Theory, applications, and computation. Hoboken, NJ: John Wiley & Sons. Inc.
   Google Scholar
• Meeker, W. Q., G. J. Hahn, and L. A. Escobar. 2017. Statistical intervals: A guide for practitioners and researchers. 2nd ed. Hoboken, NJ: John Wiley & Sons. Inc.
   Google Scholar
• Montgomery, D. C. 2013. Introduction to statistical quality control. 7th ed. Hoboken, NJ: John Wiley & Sons. Inc.
   Google Scholar
• Polansky, A. M. 1999. Upper bounds on the true coverage of bootstrap percentile type confidence intervals. The American Statistician 53 (4):362–9. doi:10.1080/00031305.1999.10474491.
   Web of Science ®Google Scholar
• Psarakis, S., A. K. Vyniou, and P. Castagliola. 2014. Some recent developments on the effects of parameter estimation on control charts. Quality and Reliability Engineering International 30 (8):1113–29. doi:10.1002/qre.1556.
   Web of Science ®Google Scholar
• Quesenberry, C. P. 1993. The effect of sample size on estimated limits for X and X control charts. Journal of Quality Technology 25 (4):237–47. doi:10.1080/00224065.1993.11979470.
   Web of Science ®Google Scholar
• Saleh, N. A., M. A. Mahmoud, M. J. Keefe, and W. H. Woodall. 2015. The difficulty in designing Shewhart X and X control charts with estimated parameters. Journal of Quality Technology 47 (2):127–38. doi:10.1080/00224065.2015.11918120.
   Web of Science ®Google Scholar
• Saleh, N. A., I. M. Zwetsloot, M. A. Mahmoud, and W. H. Woodall. 2016. CUSUM charts with controlled conditional performance under estimated parameters. Quality Engineering 28 (4):402–15. doi:10.1080/08982112.2016.1144072.
   Web of Science ®Google Scholar
• Shewhart, W. A. 1926. Quality control charts. Bell System Technical Journal 5 (4):593–603. doi:10.1002/j.1538-7305.1926.tb00125.x.
   Google Scholar
• Stigler, S. M. 1977. Fractional order statistics, with applications. Journal of the American Statistical Association 72 (359):544–50. doi:10.1080/01621459.1977.10480611.
   Web of Science ®Google Scholar
• Stuart, A., and K. Ord. 2010. Kendall’s advanced theory of statistics, distribution theory. Vol. 1. UK: John Wiley & Sons. Inc.
   Google Scholar
• TC69, ISO. 2013. Applications of statistical methods in process management: ISO 7870-2: 2013-Control charts–Part 2: Shewhart control charts. Switzerland: The International Organization for Standardization.
   Google Scholar
• Tibshirani, R. J., and B. Efron. 1993. An introduction to the bootstrap. New York, NJ: Chapman & Hall. Inc.
   Google Scholar
• Wilks, S. S. 1941. Determination of sample sizes for setting tolerance limits. The Annals of Mathematical Statistics 12 (1):91–6. doi:10.1214/aoms/1177731788.
   Google Scholar
• Wilks, S. S. 1962. Mathematical statistics. New York–London: John Wiley & Sons. Inc.
   Google Scholar
• Woodall, W. H. 2017. Bridging the gap between theory and practice in basic statistical process monitoring. ” Quality Engineering 29 (1):2–15.
   Web of Science ®Google Scholar
• Young, D. S. 2010. Tolerance: An R package for estimating tolerance intervals. Journal of Statistical Software 36 (5):1–39. doi:10.18637/jss.v036.i05.
   Web of Science ®Google Scholar
• Young, D. S., and T. Mathew. 2014. Improved nonparametric tolerance intervals based on interpolated and extrapolated order statistics. Journal of Nonparametric Statistics 26 (3):415–32. doi:10.1080/10485252.2014.906594.
   Web of Science ®Google Scholar