Skip to main content
SpringerLink
Log in

On phase II monitoring of the probability distributions of univariate continuous processes

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

Statistical process control (SPC) charts are widely used in industry for monitoring the stability of certain sequential processes like manufacturing, health care systems etc. Most SPC charts assume that the parametric form of the “in-control” process distribution \(F_1\) is available. However, it has been demonstrated in the literature that their performances are unreliable when the pre-specified process distribution is incorrect. Moreover, most SPC charts are designed to detect any shift in mean and/or variance. In real world problems, shifts in higher moments can happen without much change in mean or variance. If we fail to detect those and let the process run, it can eventually become worse and a shift in mean or variance can creep in. Moreover, the special cause that initiated the shift can inflict further damage to the system, and it may become a financial challenge to fix it. This paper provides an efficient and easy-to-use control chart for phase II monitoring of univariate continuous processes when the parametric form of the “in-control” process distribution is unknown, but Phase I observations that are believed to be i.i.d. realizations from unknown \(F_1\) are available. Data-driven practical guidelines are also provided to choose the tuning parameter and the corresponding control limit of the proposed SPC chart. Numerical simulations and a real-life data analysis show that it can be used in many practical applications.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Albers W, Kallenberg WCM (2004) Empirical nonparametric control charts: estimation effects and corrections. J Appl Stat 31:345–360

    Article  MathSciNet  MATH  Google Scholar 

  • Albers W, Kallenberg WCM (2009) CUMIN charts. Metrika. 70:111–130

    Article  MathSciNet  MATH  Google Scholar 

  • Albers W, Kallenberg WCM, Nurdiati S (2006) Data driven choice of control charts. J Stat Plan Inference 136:909–941

    Article  MathSciNet  MATH  Google Scholar 

  • Amin R, Reynolds MR Jr, Bakir ST (1995) Nonparametric quality control charts based on the sign statistic. Commun Stat-Theor Methods 24:1597–1623

    Article  MathSciNet  MATH  Google Scholar 

  • Amin RW, Searcy AJ (1991) A nonparametric exponentially weighted moving average control scheme. Commun Stat-Simul 20:1049–1072

    Article  MATH  Google Scholar 

  • Amin RW, Widmaier O (1999) Sign control charts with variable sampling intervals. Commun Stat 28:1961–1985

    Article  MATH  Google Scholar 

  • Bakir ST (2004) A distribution-free Shewhart quality control chart based on signed-ranks. Qual Eng 16:613–623

    Article  Google Scholar 

  • Bakir ST (2006) Distribution-free quality control charts based on signed-rank-like statistics. Commun Stat-Theor Methods 35:743–757

    Article  MathSciNet  MATH  Google Scholar 

  • Bakir ST, Reynolds MR Jr (1979) A nonparametric procedure for process control based on within group ranking. Technometrics 21:175–183

    Article  MATH  Google Scholar 

  • Borror CM, Montgomery DC, Runger GC (1999) Robustness of the EWMA control chart to non-normality. J Qual Technol 31:309–316

    Google Scholar 

  • Casella G, Berger R (2002) Statistical inference, 2nd edn. Duxbury, Belmont CA

    MATH  Google Scholar 

  • Chakraborti S, Eryilmaz S (2007) A nonparametric Shewhart-type signed-rank control chart based on runs. Commun Stat-Simul Comput 36:335–356

    Article  MathSciNet  MATH  Google Scholar 

  • Chakraborti S, Eryilmaz S, Human SW (2009) A phase II nonparametric control chart based on precedence statistics with runs-type signaling rules. Comput Stat Data Anal 53:1054–1065

    Article  MathSciNet  MATH  Google Scholar 

  • Chakraborti S, van der Laan P, Bakir ST (2001) Nonparametric control charts: an overview and some results. J Qual Technol 33:304–315

    Google Scholar 

  • Chakraborti S, van der Laan P, van de Wiel MA (2004) A class of distribution-free control charts. J R Stat Soc (Ser C)-Appl Stat 53:443–462

    Article  MathSciNet  MATH  Google Scholar 

  • Hackl P, Ledolter J (1992) A new nonparametric quality control technique. Commun Stat-Simul Comput 21:423–443

    Article  MathSciNet  MATH  Google Scholar 

  • Hawkins DM (1981) A CUSUM for a scale parameter. J Qual Technol 13:228–235

    Google Scholar 

  • Hawkins DM, Deng Q (2010) A nonparametric change-point control chart. J Qual Technol 42:165–173

    Google Scholar 

  • Hawkins DM, Olwell DH (1998) Cumulative sum charts and charting for quality improvement. Springer, New York

    Book  MATH  Google Scholar 

  • Hawkins DM, Qiu P, Kang CW (2003) The changepoint model for statistical process control. J Qual Technol 35:355–366

    Google Scholar 

  • Hawkins DM, Zamba KD (2005) Statistical process control for shifts in mean or variance using a changepoint formulation. Technometrics. 47:164–173

    Article  MathSciNet  Google Scholar 

  • Jones-Farmer LA, Jordan V, Champ CW (2009) Distribution-free phase I control charts for subgroup location. J Qual Technol 41:304–317

    Google Scholar 

  • Komsta L, Novomestky, F (2012) moments: Moments, cumulants, skewness, kurtosis and related tests. R package version 0.13. http://CRAN.R-project.org/package=moments

  • Liu L, Tsung F, Zhang J (2014) Adaptive nonparametric CUSUM scheme for detecting unknown shifts in location. Int J Prod Res 52:1592–1606

    Article  Google Scholar 

  • Loève M (1955) Probability theory. D. van Nostrand, New York

    MATH  Google Scholar 

  • Lucas JM, Crosier RB (1982) Robust CUSUM: a robust study for CUSUM quality control schemes. Commun Stat-Theor Methods 11:2669–2687

    Article  MathSciNet  MATH  Google Scholar 

  • Moustakides GV (1986) Optimal stopping times for detecting changes in distributions. Annal Stat 14:1379–1387

    Article  MathSciNet  MATH  Google Scholar 

  • Page ES (1954) Continuous inspection schemes. Biometrika 41:100–114

    Article  MathSciNet  MATH  Google Scholar 

  • Qiu P (2008) Distribution-free multivariate process control based on log-linear modeling. IIE Trans 40:664–677

    Article  Google Scholar 

  • Qiu P (2013) Introduction to statistical process control. CRC Press, Taylor & Francis Group, New York

    Google Scholar 

  • Qiu P, Hawkins DM (2001) A rank based multivariate CUSUM procedure. Technometrics. 43:120–132

    Article  MathSciNet  Google Scholar 

  • Qiu P, Hawkins DM (2003) A nonparametric multivariate cumulative sum procedure for detecting shifts in all directions. J R Stat Soc (Ser D) 52:151–164

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Ross GJ (2013) cpm: sequential parametric and nonparametric change detection. R package version 1.1. http://CRAN.R-project.org/package=cpm

  • Ross GJ, Adams NM (2012) Two nonparametric control charts for detecting arbitrary distribution changes. J Qual Technol 44:102–116

    Google Scholar 

  • Shewhart WA (1931) Economic control of quality of manufactured product. Van Nostrand, New York

    Google Scholar 

  • Yeh AB, Lin DKJ, Venkatramani C (2004) Unified CUSUM charts for monitoring process mean and variability. Qual Technol Quant Manag 1:65–86

    MathSciNet  Google Scholar 

  • Zeileis A, Hothorn T (2002). Diagnostic checking in regression relationships. R News 2(3), 7–10. http://CRAN.R-project.org/doc/Rnews/

  • Zhou C, Zou C, Zhang Y, Wang Z (2009) Nonparametric control chart based on change-point model. Stat Pap 50:13–28

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author thanks the associate editor and an anonymous referee for their valuable comments that significantly improved the quality of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Boise State University, Boise, ID, 83725-1555, USA

    Partha Sarathi Mukherjee

Authors
  1. Partha Sarathi Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Partha Sarathi Mukherjee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukherjee, P.S. On phase II monitoring of the probability distributions of univariate continuous processes. Stat Papers 57, 539–562 (2016). https://doi.org/10.1007/s00362-015-0668-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-015-0668-0

Keywords

Search

Navigation