Skip to main content
SpringerLink
Log in

The Use of Inequalities of Camp-Meidell Type in Nonparametric Statistical Process Monitoring

  • Chapter
Frontiers in Statistical Quality Control 11

Part of the book series: Frontiers in Statistical Quality Control ((FSQC))

Abstract

A few authors have used the classical Camp-Meidell inequality for the nonparametric analysis of statistical process monitoring. The following issues have not received sufficient attention. (i) The use of moments of order higher than 2 in the inequalities provides tighter bounds. (ii) The problem of estimating the moments in the bounds, e.g., from a phase 1 sample, cannot be neglected. The present study analyses both aspects (i) and (ii). Appropriate estimators, their properties, and the effect of estimation on the properties of process monitoring charts are investigated. In particular, the use of empirical Camp-Meidell bounds in quantile control charts is studied.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Amin, R. W., Reynolds, M. R., & Saad, B. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistics – Theory and Methods, 24(6), 1597–1623.

    Article  MATH  MathSciNet  Google Scholar 

  • Areepong, Y., & Sukparungsee, S. (2010). An integral equation approach to EWMA chart for detecting a change in lognormal distribution. Thailand Statistician, 8(1), 47–61.

    Google Scholar 

  • Bai, D. S., & Choi, I. S. (1995). \(\bar{X}\) and R control charts for skewed populations. Journal Of Quality Technology, 27, 120–131.

    Google Scholar 

  • Bakir, S. T. (2004). A distribution-free Shewhart quality control charts based on signed-ranks. Quality Engineering, 16(4), 613–623.

    Article  Google Scholar 

  • Bakir, S. T. (2006). Distribution-free quality control charts based on signed-rank-like statistics. Communications in Statistics – Theory and Methods, 35(4), 743–757.

    Article  MATH  MathSciNet  Google Scholar 

  • Bakir, S. T., & Reynolds, M. R. (1979). A nonparametric procedure for process control based on within-group ranking. Technometrics, 21(2), 175–183.

    Article  MATH  Google Scholar 

  • Barndorff-Nielsen, O.E., & Blæsild, P. (1981). Hyperbolic distributions and ramifications: Contributions to theory and application. In C. Taillie, G. Patil, & B. Baldessari (Eds.), Statistical distributions in scientific work (Vol. 4, pp. 19–44). Dordrecht: Reidel.

    Chapter  Google Scholar 

  • Betul, K., & Yaziki, B. (2006). The individual control charts for BURR distributed data. In Proceedings of the Ninth WSEAS International Conference on Applied Mathematics (pp. 645–649). Istanbul, Turkey.

    Google Scholar 

  • Burr, I. W. (1967). The effects of non-normality on constants for \(\bar{X}\) and R charts. Industrial Quality Control, 24, 563–569.

    Google Scholar 

  • Camp, B. H. (1922). A new generalization of Tchebycheff’s inequality. Bulletin of the American Mathematical Society, 28, 427–432.

    Article  MATH  MathSciNet  Google Scholar 

  • Chakraborti, S., & Eryilmaz, S. (2007). A nonparametric Shewhart-type signed-rank control chart based on runs. Communications in Statistics – Simulation and Computation, 36(2), 335–356.

    Article  MATH  MathSciNet  Google Scholar 

  • Chakraborti, S., van der Laan, P., & Bakir, S. T. (2001). Nonparametric control charts: An overview and some results. Journal Of Quality Technology, 33(3), 304–315.

    Google Scholar 

  • Chan, L. K., & Cui, H. J. (2003). Skewness correction \(\bar{X}\) and R charts for skewed distributions. Naval Research Logistics, 50, 1–19.

    Article  MathSciNet  Google Scholar 

  • Chang, Y. S., & Bai, D. S. (2001). Control charts for positively skewed populations with weighted standard deviations. Quality and Reliability Engineering International, 17(5), 397–406.

    Article  Google Scholar 

  • Chatterjee, S., & Qiu, P. (2009). Distribution-free cumulative sum control charts using bootstrap-based control limits. The Annals of Applied Statistics, 3(1), 349–369.

    Article  MATH  MathSciNet  Google Scholar 

  • Chebyshev, B. L. (1867). Des valeurs moyennes. Journal de Mathématique Pure et Appliquée, 12, 177–184.

    Google Scholar 

  • Chen, G. (1998). An improved p chart through simple adjustments. Journal of Quality Technology, 30, 142–151.

    Google Scholar 

  • Cheng, S. W., & Xie, H. (2000). Control charts for lognormal data. Tamkang Journal of Science and Engineering, 3, 131–137.

    Google Scholar 

  • Choobineh, F., & Ballard, J. L. (1987). Control-limits of QC charts for skewed distributions using weighted variance. IEEE Transactions on Reliability, 36, 473–477.

    Article  MATH  Google Scholar 

  • Cornish, E. A., & Fisher, R. A. (1938). Moments and cumulants in the specifications of distributions. Revue de l’Institut Internationale de Statistique, 5(4), 307–320.

    Article  Google Scholar 

  • Cramér, H. (1946). Mathematical methods of statistics. Princeton, NJ, Princeton University Press.

    MATH  Google Scholar 

  • Derya, K., & Canan, H. (2012). Control charts for skewed distributions: Weibull, gamma, and lognormal. Metodolos̆ki zvezki, 9(2), 95–106.

    Google Scholar 

  • Edgeman, R. L. (1989). Inverse Gaussian control charts. Australian Journal of Statistics, 31(1), 435–446.

    Article  MathSciNet  Google Scholar 

  • Erto, P., & Pallotta, G. (2007). A new control chart for Weibull technological processes. Quality Technology and Quantitative Management, 4(4), 553–567.

    MathSciNet  Google Scholar 

  • Erto, P., Pallotta, G., & Park, S. H. (2008). An example of Data Technology product: A control chart for Weibull processes. International Statistical Review, 76(2), 157–166.

    Article  Google Scholar 

  • Ferrell, E. B. (1958). Control charts for lognormal universe. Industrial Quality Control, 15, 4–6.

    Google Scholar 

  • Fisher, R. A. (1928). Moments and product moments of sampling distributions. Proceedings of the London Mathematical Society, 30, 199–238.

    Google Scholar 

  • Florac, W. A., & Carleton, A. D. (1999). Measuring the software process: Statistical process control for software process improvement. Boston: Addison-Wesley.

    Google Scholar 

  • Göb, R., & Lurz, K. (2013a). Estimating the fourth central moment for unimodal distributions. Preprints of the Institute of Mathematics, 334. Würzburg: University of Würzburg.

    Google Scholar 

  • Göb R, Lurz K (2013b). Empirical quantile bounds based on the Camp-Meidell inequality. Preprints of the Institute of Mathematics, 335. Würzburg: University of Würzburg.

    Google Scholar 

  • González, I., & Viles, E. (2000). Semi-economic design of X-control charts assuming Gamma distribution. Economic Quality Control, 15, 109–118.

    MATH  MathSciNet  Google Scholar 

  • González, I., & Viles, E. (2001). Design of R control chart assuming a Gamma distribution. Economic Quality Control, 16(2), 199–204.

    Article  MATH  MathSciNet  Google Scholar 

  • Grant, E. L., & Leavenworth, R. S. (1999). Statistical Quality Control 7th Ed. Boston: McGraw-Hill.

    Google Scholar 

  • Guo, B. C., & Wang, B. X. (2012). Control charts For monitoring the Weibull Shape parameter based on type-II censored sample. Quality and Reliability Engineering International, 30, 13–24. http://dx.doi.org/10.1002/qre.1473

  • Haas, M., & Pigorsch, C. (2009). Financial economics, fat-tailed distributions. In R. A. Meyers (Ed.) Encyclopedia of Complexity and Systems Science (pp. 3404–3435). New York: Springer.

    Chapter  Google Scholar 

  • Hackl, P., & Ledolter, J. (1991). A control chart based on ranks. Journal of Quality Technology, 23(2), 117–124.

    Google Scholar 

  • Hackl, P., & Ledolter, J. (1992). A new nonparametric quality control technique. Communications in Statistics – Simulation and Computation, 21(2), 423–443.

    Article  MATH  MathSciNet  Google Scholar 

  • Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. American Statistician, 50, 361–365.

    Google Scholar 

  • International Organisation for Standardisation (ISO) (2008). SO/IEC Guide 98–3:2008: Uncertainty of measurement - Part 3: Guide to the expression of uncertainty in measurement. ISO: Geneva, Switzerland.

    Google Scholar 

  • Ion, R. A. (2001). Nonparametric Statistical Process Control. Korteweg-de Vries Instituut voor Wiskunde, Faculteit der Natuurwetenschappen, Wiskunde en Informatica. Amsterdam: Universiteit van Amsterdam.

    Google Scholar 

  • Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47(1), 183–189.

    Google Scholar 

  • Joffe, A. D., & Sichel, H. S. (1968). A chart for sequentially testing observed arithmetic means from lognormal populations against a given standard. Technometrics, 10, 605–612.

    Article  Google Scholar 

  • Johnson, N. L. (1966). Cumulative sum control charts and the Weibull distribution. Technometrics, 8(3), 481–491.

    Article  MathSciNet  Google Scholar 

  • Jones, L. A., & Woodall, W. H. (1998). The performance of bootstrap control charts. Journal of Quality Technology, 30, 362–375.

    Google Scholar 

  • Kanji, G. K., & Arif, O. H. (2001). Median rankit control chart for Weibull distribution. Total Quality Management, 12(5), 629–642.

    Google Scholar 

  • Kantam, R. R. L., & Sriram, B. (2001). Variable control charts based on Gamma distribution. IAPQR Transactions, 26(2), 63–67.

    MATH  MathSciNet  Google Scholar 

  • Kantam, R. R. L., Vasudeva, R. A., & Srinivasa, R. G. (2006). Control charts for log-logistic distribution. Economic Quality Control, 21(1), 77–86.

    Article  MATH  Google Scholar 

  • Kotz, S., & Lovelace, C. R. (1988). Process capability indices in theory and practice. London: Edward Arnold.

    Google Scholar 

  • Leger, C., Politis, D., & Romano, J. (1992). Bootstrap technology applications. Technometrics, 34, 378–398.

    Article  MATH  Google Scholar 

  • Liu, R. Y., & Tang, J. (1996). Control charts for dependent and independent measurements based on bootstrap methods. Journal of the American Statistical Association, 91, 1694–1700.

    Article  MATH  MathSciNet  Google Scholar 

  • Meidell, M. B. (1922). Sur un problème du calcul des probabilités et les statistiques mathématiques. Comptes Rendus, 175, 806–808.

    Google Scholar 

  • Morrison, J. (1958). The lognormal distribution in quality control. Applied Statistics, 7, 160–172.

    Article  Google Scholar 

  • Necula, C. (2009). Modelling heavy-tailed stock index returns using the generalised hyperbolic distribution. Romanian Journal of Economic Forecasting, 2, 118–131.

    Google Scholar 

  • Nelson, P. R. (1979). Control charts for Weibull processes with standards given. IEEE Transactions on Reliability, 28, 383–387.

    Google Scholar 

  • Park, C., & Reynolds, M. R. (1987). Nonparametric procedures for monitoring a location parameter based on linear placement statistics. Sequential Analysis, 6(4), 303–323.

    Article  MATH  MathSciNet  Google Scholar 

  • Patel, J. K., Kapadia, C. H., & Owen, D. B. (1976). Handbook of statistical distributions. New York/Basel: Marcel Dekker Inc.

    MATH  Google Scholar 

  • Pyzdek, I. (1995). Why normal distributions aren’t [all that normal]. Quality Engineering, 7(4), 769–777.

    Article  Google Scholar 

  • Qiu, P., & Li, Z. (2011). On nonparametric statistical process control of univariate processes. Technometrics, 53(4), 390–405.

    Article  MathSciNet  Google Scholar 

  • Ramalhoto, M. F., & Morais, M. (1998). EWMA control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals. Economic Quality Control, 13, 23–46.

    MATH  Google Scholar 

  • Ramalhoto, M. F., & Morais, M. (1999). Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals. Journal of Applied Statistics, 26(1), 129–160.

    Google Scholar 

  • Samanta, B., & Bhattacherjee, A. (2004). Problem of nonnormality in statistical quality control. Journal of the South African Institute of Mining and Metallurgy, 104, 257–264.

    Google Scholar 

  • Schilling, E. G., & Nelson, P. R. (1976). The effect of nonnormality on the control limits of charts. Journal of Quality Technology, 8(4), 179–193.

    Google Scholar 

  • Seppala, T., Moskowitz, H., Plante, R., & Tang, J. (1995). Statistical process control via the subgroup bootstrap. Journal of Quality Technology, 27, 139–153.

    Google Scholar 

  • Shewhart, W. A. (1931). Economic control of quality of manufactured product. London: Macmillan.

    Google Scholar 

  • Spedding, T. A., & Rawlings, P. L. (1994). Nonnormality in statistical process control measurements. International Journal of Quality & Reliability Management, 11(6), 27–37.

    Google Scholar 

  • Subba, R. R., & Kantam, R. R. L. (2008). Variable control charts for process mean with double exponential distribution. Acta Cinica Indica, 34(4), 1925–1930.

    Google Scholar 

  • Teyarachakul, S., Chand, S., & Tang, J. (2007). Estimating the limits for statistical process control charts: A direct method improving upon the bootstrap. European Journal of Operational Research, 178, 472–481.

    Article  MATH  MathSciNet  Google Scholar 

  • Willemain, T. R., & Runger, G. C. (1996). Designing control charts using an empirical reference distribution. Journal of Quality Technology, 28(1), 31–38.

    Google Scholar 

  • Winterbottom, A. (1993). Simple adjustments to improve control limits on attribute charts. Quality and Reliability Engineering International, 9, 105–109.

    Article  Google Scholar 

  • Woodall, W. H. (2000). Controversies and contradictions in statistical process control. Journal of Quality Technology, 32(4), 341–350.

    Google Scholar 

  • Wu, C., & Wang, Z. (2009). A synthetic control chart based on the Cornish-Fisher expansion. Chinese Journal of Applied Probability and Statistics, 25(3), 258–265.

    Google Scholar 

  • Wu, Z. (1996). Asymmetric control limits of the X-bar chart for skewed process distributions. International Journal of Quality & Reliability Management, 13(9), 49–60.

    Article  Google Scholar 

  • Yang, Z., & Xie, M. (2000). Process monitoring of exponentially distributed characteristics through an optimal normalizing transformation. Journal of Applied Statistics, 27, 1051–1063.

    Article  MATH  Google Scholar 

  • Yourstone, S. A., & Zimmer, W. J. (1992). Nonnormality and the design of control charts for averages. Decision Sciences, 23, 1099–1113.

    Article  Google Scholar 

  • Zhang, L., & Chen, G. (2004). EWMA charts for monitoring the mean of censored Weibull lifetimes. Journal of Quality Technology, 36, 321–328.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics and Statistics, University of Würzburg, Sanderring 2, 97070, Würzburg, Germany

    Rainer Göb & Kristina Lurz

Authors
  1. Rainer Göb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kristina Lurz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rainer Göb .

Editor information

Editors and Affiliations

  1. Helmut Schmidt University, Hamburg, Germany

    Sven Knoth

  2. Europa-Universität Viadrina, Frankfurt/Oder, Brandenburg, Germany

    Wolfgang Schmid

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Göb, R., Lurz, K. (2015). The Use of Inequalities of Camp-Meidell Type in Nonparametric Statistical Process Monitoring. In: Knoth, S., Schmid, W. (eds) Frontiers in Statistical Quality Control 11. Frontiers in Statistical Quality Control. Springer, Cham. https://doi.org/10.1007/978-3-319-12355-4_11

Download citation

Publish with us

Policies and ethics

Search

Navigation