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

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 $P(N > 0), P(N > 1),\ldots,$ it is shown that $(m^-_n)^iP(N > n) \leqslant P(N > n + i) \leqslant (m^+_n)^iP(N > n)$ for...

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Bibliographic Details
Published in:Applied Statistics 1986-01, Vol.35 (2), p.151-158
Main Author: Waldmann, K.‐H.
Format: Article
Language:English
Subjects:
Applications
Applied statistics
Approximation
Average run length
Distribution functions
Exact sciences and technology
Extrapolation methods
Geometric moving average chart
Mathematical extrapolation
Mathematics
Probability and statistics
Probability mass distributions
Quality assurance
Quality control
Random variables
Run‐length distribution
Sciences and techniques of general use
Simpsons rule
Statistics
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Summary: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 $P(N > 0), P(N > 1),\ldots,$ it is shown that $(m^-_n)^iP(N > n) \leqslant P(N > n + i) \leqslant (m^+_n)^iP(N > n)$ for each n and all i = 1, 2,... with constants 0 ⩽ m-n ⩽ m+n ⩽ 1 suitably chosen. The bounds converge monotonically and, under some mild and natural assumptions, m-n and m+n 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.
ISSN:0035-9254
1467-9876
DOI:10.2307/2347265