The Annals of Applied Statistics

Distribution-free cumulative sum control charts using bootstrap-based control limits

Snigdhansu Chatterjee and Peihua Qiu

Full-text: Open access


This paper deals with phase II, univariate, statistical process control when a set of in-control data is available, and when both the in-control and out-of-control distributions of the process are unknown. Existing process control techniques typically require substantial knowledge about the in-control and out-of-control distributions of the process, which is often difficult to obtain in practice. We propose (a) using a sequence of control limits for the cumulative sum (CUSUM) control charts, where the control limits are determined by the conditional distribution of the CUSUM statistic given the last time it was zero, and (b) estimating the control limits by bootstrap. Traditionally, the CUSUM control chart uses a single control limit, which is obtained under the assumption that the in-control and out-of-control distributions of the process are Normal. When the normality assumption is not valid, which is often true in applications, the actual in-control average run length, defined to be the expected time duration before the control chart signals a process change, is quite different from the nominal in-control average run length. This limitation is mostly eliminated in the proposed procedure, which is distribution-free and robust against different choices of the in-control and out-of-control distributions.

Article information

Ann. Appl. Stat., Volume 3, Number 1 (2009), 349-369.

First available in Project Euclid: 16 April 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Cumulative sum control charts distribution-free procedures nonparametric model statistical process control resampling robustness


Chatterjee, Snigdhansu; Qiu, Peihua. Distribution-free cumulative sum control charts using bootstrap-based control limits. Ann. Appl. Stat. 3 (2009), no. 1, 349--369. doi:10.1214/08-AOAS197.

Export citation


  • Bajgier, S. M. (1992). The use of bootstrapping to construct limits on control charts. In Proc. Decision Science Inst., San Diego, 1611–1613.
  • Bakir, S. T. and Reynolds, M. R. Jr. (1979). A nonparametric procedure for process control based on within group ranking. Technometrics 21 175–183.
  • Bolton, R. J. and Hand, D. J. (2002). Statistical fraud detection: A review. Statist. Sci. 17 235–255.
  • Chakraborti, S., van Der Laan, P. and Bakir, S. T. (2001). Nonparametric control charts: An overview and some results. J. Quality Technology 33 304–315.
  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1–26.
  • Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall, Boca Raton, FL.
  • Falk, M. and Reiss, R. D. (1989). Weak convergence of smoothed and nonsmoothed bootstrap quantile estimates. Ann. Probab. 17 362–371.
  • Franklin, L. A. and Wasserman, G. S. (1992). Bootstrap lower confidence limits for capability indices. J. Quality Technology 24 196–210.
  • Hall, P., DiCiccio, T. J. and Romano, J. P. (1989). On smoothing and the bootstrap. Ann. Statist. 17 692–704.
  • Hawkins, D. M. and Olwell, D. H. (1998). Cumulative Sum Charts and Charting for Quality Improvement. Springer, New York.
  • Jones, L. A., Champ, C. W. and Rigdon, S. E. (2004). The run length distribution of the CUSUM with estimated parameters. J. Quality Technology 36 95–108.
  • Krawczak, M., Ball, E., Fenton, I., Stenson, P., Abeysinghe, S., Thomas, N. and Cooper, D. N. (1999). Human gene mutation database: A biomedical information and research resource. Human Mutation 15 45–51.
  • Liu, R. Y. and Tang, J. (1996). Control charts for dependent and independent measurements based on bootstrap methods. J. Amer. Statist. Assoc. 91 1694–1700.
  • Lorden, G. (1971). Procedures for reacting to a change in distribution. Ann. Math. Statist. 42 1897–1908.
  • Lu, C. W. and Reynolds, M. R., Jr. (1999). Control charts for monitoring the mean and variance of autocorrelated processes. J. Quality Technology 31 259–274.
  • Meulbroek, L. K. (1992). An empirical analysis of illegal insider trading. J. Finance 47 1661–1699.
  • Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist. 14 1379–1387.
  • Page, E. S. (1954). Continuous inspection schemes. Biometrika 41 100–114.
  • Qiu, P. (2008). Distribution-free multivariate process control based on log-linear modeling. IIE Transactions 40 664–677.
  • Qiu, P. and Hawkins, D. (2001). A rank based multivariate CUSUM procedure. Technometrics 43 120–132.
  • Qiu, P. and Hawkins, D. (2003). A nonparametric multivariate CUSUM procedure for detecting shifts in all directions. J. Roy. Statist. Soc. Ser. D 52 151–164.
  • Reynolds, M. R. Jr. (1975). Approximations to the average run length in cumulative sum control charts. Technometrics 17 65–71.
  • Scariano, S. and Hebert, J. (2003). Adapting EWMA control charts for batch-correlated data. Quality Engineering 15 545–556.
  • Seppala, T., Moskowitz, H., Plante, R. and Tang, J. (1995). Statistical process control via the subgroup bootstrap. J. Quality Technology 27 139–153.
  • Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer, New York.
  • Shapiro, S. S. and Wilk, M. B. (1965). An analysis of variance test for normality: Complete samples. Biometrika 52 591–611.
  • Siegmund, D. (1985). Sequential Analysis. Springer, New York.
  • Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
  • Steiner, S. H. (1999). EWMA control charts with time-varying control limits and fast initial response. J. Quality Technology 31 75–86.
  • Steiner, S. H., Cook R. and Farewell, V. (1999). Monitoring paired binary surgical outcomes using cumulative sum charts. Statistics in Medicine 8 69–86.
  • Willemain, T. R. and Runger, G. C. (1996). Designing control charts using an empirical reference distribution. J. Quality Technology 28 31–38.
  • Wood, M., Kaye, M. and Capon, N. (1999). The use of resampling for estimating control chart limits. J. Operational Res. Soc. 50 651–659.
  • Wu, Z. and Wang, Q. (1996). Bootstrap control charts. Quality Engineering 9 143–150.
  • Zhang, N. F. (1998). A statistical control chart for stationary process data. Technometrics 40 24–38.