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Statistische Hefte

, 27:37 | Cite as

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

  • K. H. Waldmann
Articles

Summary

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.

Key-words

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 

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

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

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