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Bounds for the Distribution of the Run Length of Geometric Moving Average Charts

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  • K.‐H. Waldmann

Abstract

Upper and lower bounds are derived for the distribution of the run length N of one‐sided and two‐sided geometric moving average charts. By considering the iterates p(N > 0), p(N > 1), ..., it is shown that (mn−)ip(N > n) ≤ p(N > n + i) ≤ (mn+)i p(N > n) for each n and all i = 1, 2, ... with constants 0 ≤ mn− ≤ mn+ ≤ 1 suitably chosen. The bounds converge monotonically and, under some mild and natural assumptions, mn− and mn+ have the same positive limit as n → ∞. Bounds are also presented for the percentage points of the distribution function of N, for the first two moments of N, and for the probability mass function of N. Some numerical results are displayed to demonstrate the efficiency of the method.

Suggested Citation

  • K.‐H. Waldmann, 1986. "Bounds for the Distribution of the Run Length of Geometric Moving Average Charts," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 35(2), pages 151-158, June.
    • Handle: RePEc:bla:jorssc:v:35:y:1986:i:2:p:151-158
    DOI: 10.2307/2347265
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    Cited by:

    1. Sánchez, Ismael, 2015. "Adaptive EWMA Control Charts with a Time Varying Smoothing Parameter," DES - Working Papers. Statistics and Econometrics. WS ws1507, Universidad Carlos III de Madrid. Departamento de Estadística.
    2. Chao-Yu Chou & Chung-Ho Chen & Hui-Rong Liu, 2006. "Economic Design of EWMA Charts with Variable Sampling Intervals," Quality & Quantity: International Journal of Methodology, Springer, vol. 40(6), pages 879-896, December.
    3. Yu-min Liu & Li Xue, 2015. "The optimization design of EWMA charts for monitoring environmental performance," Annals of Operations Research, Springer, vol. 228(1), pages 113-124, May.
    4. Margavio, Thomas M. & Conerly, Michael D. & Woodall, William H. & Drake, Laurel G., 1995. "Alarm rates for quality control charts," Statistics & Probability Letters, Elsevier, vol. 24(3), pages 219-224, August.
    5. Seven Knoth, 2005. "Fast initial response features for EWMA control charts," Statistical Papers, Springer, vol. 46(1), pages 47-64, January.

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