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Bounds for the Distribution of the Run Length of Geometric Moving Average Charts
Author(s) -
Waldmann K.H.
Publication year - 1986
Publication title -
journal of the royal statistical society: series c (applied statistics)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.205
H-Index - 72
eISSN - 1467-9876
pISSN - 0035-9254
DOI - 10.2307/2347265
Subject(s) - statistics , mathematics , geometric distribution , distribution (mathematics) , probability distribution , mathematical analysis
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), ..., it is shown that ( m n − ) i p ( N > n ) ≤ p ( N > n + i ) ≤ ( m n + ) i p ( 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.