A general framework is provided for detecting a change in the distribution of sequentially observed random variables. The first stage, N, at which such a change is signaled, is a random variable whose distribution measures the performance of the procedure. Based on P(N>0), P(N>1), ..., P(N>n), n∈IN, bounds are constructed for P(N>n+i) such that (mn−)iP(N>n)≤P(N>n+i)≤(mn+)iP(N>n) holds for all i∈IN with suitable constants 0≤mn−≤mn+≤1. The bounds monotonically converge in the sense that ±(mn±)i+1P(N>n)≥±(mn+1±)iP(N>n+1), and, under some mild and natural assumption, lim mn−=lim mn+ > 0. Some numerical results are displayed for CUSUM control charts to demonstrate the efficiency of the method.
inspection schemes control charts average run length run-length distribution cumulative sum (CUSUM) control charts moving average (MOSUM) charts geometric moving average charts extrapolation methods
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