Bounds for the Distribution of the Run Length of One-Sided and Two-Sided CUSUM Quality Control Schemes

Abstract

Upper and lower bounds are derived for the distribution of the run length N of both the one-sided and two-sided CUSUM schemes. Based upon a sequence of iterates Pr(N > 0), Pr(N > 1), …, bounds are constructed such that (m n −) i Pr(N > n) ≤ Pr(N > n + i) ≤ (m n +) i Pr(N > n) holds for all n, i = 1, 2,…, with constants 0 ≤ m n −; ≤ m n + ≤ 1 suitably chosen. The bounds converge monotonically in the sense that (m n −) i+1Pr(N > n) ≤ (m n+1 −) i Pr(N > n + 1), (m n +) i+1Pr(N > n) ≥ (m n+1 +) i Pr(N > n + 1), and, under some mild and natural conditions, . As a by-product, bounds are presented for the percentage points of the distribution function of N, for the average run length of the inspection scheme, for the standard deviation of N, and finally, for the probability mass function of N. Some numerical results are displayed to demonstrate the effkiency of the method.

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