Bounds to the distribution of the run length in general quality-control schemes

Summary

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 (m ⁻_n )ⁱP(N>n)≤P(N>n+i)≤(m ⁺_n )ⁱP(N>n) holds for all i∈IN with suitable constants 0≤m ⁻_n ≤m ⁺_n ≤1. The bounds monotonically converge in the sense that ±(m ^±_n )ⁱ⁺¹P(N>n)≥±(m ^±_n+1 )ⁱP(N>n+1), and, under some mild and natural assumption, lim m ⁻_n =lim m ⁺_n > 0. Some numerical results are displayed for CUSUM control charts to demonstrate the efficiency of the method.