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

K.‐H. Waldmann

Journal of the Royal Statistical Society Series C, 1986, vol. 35, issue 2, 151-158

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.

Date: 1986
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