Statistische Hefte

, 27:37 | Cite as

Bounds to the distribution of the run length in general quality-control schemes

  • K. H. Waldmann


A general framework is provided for detecting a change in the distribution of sequentially observed random variables. The first stage, N, at which such a change is signaled, is a random variable whose distribution measures the performance of the procedure. Based on P(N>0), P(N>1), ..., P(N>n), n∈IN, bounds are constructed for P(N>n+i) such that (m n )iP(N>n)≤P(N>n+i)≤(m n + )iP(N>n) holds for all i∈IN with suitable constants 0≤m n ≤m n + ≤1. The bounds monotonically converge in the sense that ±(m n ± )i+1P(N>n)≥±(m n+1 ± )iP(N>n+1), and, under some mild and natural assumption, lim m n =lim m n + > 0. Some numerical results are displayed for CUSUM control charts to demonstrate the efficiency of the method.


inspection schemes control charts average run length run-length distribution cumulative sum (CUSUM) control charts moving average (MOSUM) charts geometric moving average charts extrapolation methods 


  1. 1.
    BROOK, D., and EVANS, D.A. (1972). An Approach to the Probability Distribution of CUSUM Run Length. Biometrika, 59, 539–549.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    VAN DOBBEN DE BRUYN, C.S. (1968). Cumulative Sum Tests: Theory and Practice. Griffin's Statistical Monographs and Courses, No. 24, London.Google Scholar
  3. 3.
    HARRIS, T. E. (1963). The Theory of Branching Processes. Englewood Cliffs, N. J. Prentice-Hall, Inc.MATHGoogle Scholar
  4. 4.
    KARLIN, S. and TAYLOR, H.M. (1975). A First Course in Stochastic Processes (second edition). Academic Press, New York, San Francisco, London.MATHGoogle Scholar
  5. 5.
    LAI, T.L. (1974). Control Charts based on Weighted Sums. Annals of Statistics, 2, 134–147.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    LUCAS, J.M. and CROSIER, R.B. (1982). Fast Initial Response for CUSUM Quality Control Schemes. Technometrics, 24, 199–205.CrossRefGoogle Scholar
  7. 7.
    PAGE, E.S. (1954). Continuous Inspection Schemes. Biometrika, 41, 100–114.MATHMathSciNetGoogle Scholar
  8. 8.
    ROBERTS, S.W. (1959). Control Chart Tests based on Geometric Moving Averages. Technometrics, 1, 239–250.CrossRefGoogle Scholar
  9. 9.
    ROBERTS, S.W. (1966). A Comparison of some Control Chart Procedures. Technometrics, 8, 411–430.CrossRefMathSciNetGoogle Scholar
  10. 10.
    ROBINSON, P.B. and HO, T.Y. (1978). Average Run Lengths of Geometric Moving Averages by numerical Methods. Technometrics, 20, 85–93.MATHCrossRefGoogle Scholar
  11. 11.
    SCHELLHAAS, H. (1974). Zur Extrapolation in Markoff' schen Entscheidungsmodellen mit Diskontierung. Zeitschrift für Operations Research, 18, 91–104.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    SHEWHART, W. (1931). Economic Control of Quality of Manufactured Product. Van Nostrand, Princeton.Google Scholar
  13. 13.
    WALDMANN, K.-H. (1985a). Bounds for the Distribution of the Run Length of One-Sided and Two-Sided CUSUM Quality Control Schemes. Technometrics, 27, to appear.Google Scholar
  14. 14.
    WALDMANN, K.-H. (1985b). Bounds for the Distribution of the Run Length of Geometric Moving Average Charts. Applied Statistics, to appear.Google Scholar
  15. 15.
    WALDMANN, K.-H. (1985c). On Bounds for Dynamic Programs. Mathematics of Operations Research, 10, 220–232.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    WOODALL, W.H. (1983). The Distribution of the Run Length of One-Sided CUSUM Procedures for Continuous Random Variables. Technometrics, 25, 295–301.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • K. H. Waldmann
    • 1
  1. 1.Institut für Quantitative Ökonomik und StatistikFreie Universität BerlinBerlin 33

Personalised recommendations