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Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart

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Abstract

A simple algorithm is introduced for computing the run length distribution of a monitoring scheme combining a Shewhart chart with an Exponentially Weighted Moving Average control chart. The algorithm is based on the numerical approximation of the integral equations and integral recurrence relations related to the run-length distribution. In particular, a Clenshaw-Curtis product-integration rule is applied for handling discontinuities in the integrand function due to the simultaneous use of the two control schemes. The proposed algorithm, implemented in R and publicy available, compares favourably with the Markov chain approach originally used to approximate the run length properties of the combined Shewhart-EWMA.

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References

  • Brook, D., Evans, D.: An approach to the probability distribution of CUSUM run length. Biometrika 59, 539–549 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  • Capizzi, G., Masarotto, G.: An adaptive Exponentially Weighted Moving Average control chart. Technometrics 45, 199–207 (2003)

    Article  MathSciNet  Google Scholar 

  • Champ, C.W., Rigdon, S.E.: A comparison of the Markov chain and integral equation approaches for evaluating the run length distribution of quality control charts. Commun. Stat. Simul. Comput. 20, 191–204 (1991)

    Article  MATH  Google Scholar 

  • Champ, C.W., Rigdon, S.E., Scharnag, K.A.: Method for deriving integral equations useful in control chart performance analysis. Nonlinear Anal. Theory Methods Appl. 47, 2089–2101 (2001)

    Article  MATH  Google Scholar 

  • Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  • Crowder, S.V.: A simple method for studying run-length distributions of Exponentially Weighted Moving Average charts. Technometrics 29, 401–407 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  • Imhof, J.P.: On the method for numerical integration of Clenshaw and Curtis. Numer. Math. 5, 138–141 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  • Kang, S.H., Koltracht, I., Rawitscher, G.: Nystrom-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels. Math. Comput. 72, 729–756 (2002)

    Article  MathSciNet  Google Scholar 

  • Knoth, S.: spc: Statistical Process Control (2004). R package version 0.2

  • Knoth, S.: Accurate ARL computation for EWMA-s 2 control charts. Stat. Comput. 15, 341–352 (2005)

    Article  MathSciNet  Google Scholar 

  • Lucas, J.M., Saccucci, M.S.: Exponentially weighted moving average control schemes: Properties and enhancements quality. Technometrics 32, 1–29 (1990)

    Article  MathSciNet  Google Scholar 

  • Luceno, A., Puig-Pey, J.: Evaluation of the run-length probability distribution for CUSUM charts. Technometrics 42, 411–416 (2000)

    Article  Google Scholar 

  • Luceno, A., Puig-Pey, J.: An accurate algorithm to compute the run length probability distribution, and its convolutions, for a CUSUM chart to control normal mean. Comput. Stat. Data Anal. 38, 249–261 (2002a)

    Article  MATH  MathSciNet  Google Scholar 

  • Luceno, A., Puig-Pey, J.: Computing the run length probability distribution for CUSUM charts. J. Qual. Technol. 34, 209–215 (2002b)

    Google Scholar 

  • Montgomery, D.C.: Introduction to Statistical Quality Control, 5th edn. Wiley, New York (2004)

    Google Scholar 

  • Nelder, J.A., Mead, R.: A simplex algorithm for function minimization. Comput. J. 7, 308–313 (1965)

    MATH  Google Scholar 

  • Piessens, R.: Computing integral transforms and solving integral equations using Chebyshev polynomial approximations. J. Comput. Appl. Math. 121, 113–124 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Piessens, R., de Doncker-Kapenga, E., Uberhuber, C.W., Kahaner, D.K.: QUADPACK. A Subroutine Package for Automatic Integration. Springer, Berlin (1983)

    MATH  Google Scholar 

  • R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2008). URL http://www.R-project.org

  • Shu, L.: An adaptive Exponentially Weighted Moving Average control chart for monitoring process variances. J. Stat. Comput. Simul. 30, 415–428 (2008)

    Google Scholar 

  • Sloan, I.H.: Quadrature methods for integral equations of the second kind over infinite intervals. Math. Comput. 36, 511–523 (1978)

    MathSciNet  Google Scholar 

  • Sloan, I.H.: Analysis of general quadrature methods for integral equations of the second kind. Numer. Math. 38, 263–278 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  • Sloan, I.H., Smith, W.E.: Product-integration with the Clenshaw and Curtis and related points. Numer. Math. 30, 415–428 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • Sloan, I.H., Smith, W.E.: Product integration with the Clenshaw and Curtis points: implementation and error points. Numer. Math. 34, 387–401 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  • Sloan, I.H., Smith, W.E.: Properties of interpolatory product integration rules. SIAM J. Numer. Anal. 19, 427–442 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  • Trefethen, L.: Is Gaussian quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67–87 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Statistical Sciences, University of Padua, via Cesare Battisti 241/243, 35121, Padova, Italy

    Giovanna Capizzi & Guido Masarotto

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  1. Giovanna Capizzi
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  2. Guido Masarotto
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Correspondence to Giovanna Capizzi.

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This research was partially funded by Italian MIUR-Cofin 2006 grants.

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Capizzi, G., Masarotto, G. Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart. Stat Comput 20, 23–33 (2010). https://doi.org/10.1007/s11222-008-9113-8

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  • DOI: https://doi.org/10.1007/s11222-008-9113-8

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