A lower confidence bound for the change point after a sequential CUSUM test

Abstract

A lower confidence bound for the change point after a CUSUM test is proposed by tracing back a predetermined number of zero points of the monitoring CUSUM process before the detection. Due to the renewal property of the CUSUM process, all related quantities can be computed with fixed memory. By assuming that the change point is far from the beginning and the in-control average run length of samples is large, the non-coverage probability and average length from the lower confidence bound to detection time are derived by conditioning on detection. Furthermore, local second order expansions for the related quantities are obtained and shown by simulation to be quite accurate in the case of detecting an increase in mean or variance when sampling from a normal distribution.