Design of Cumulative Sum Schemes for Start-Up Processes and Short Runs

Abstract

A cumulative sum (CUSUM) of Q statistics was previously proposed for detecting shifts in the process mean when the parameters of the process distribution are unknown, such as in start-up processes and short runs. This CUSUM scheme is modeled on the classic CUSUM procedure for detecting a shift in the mean of a normally distributed process. The design constants derived to give the classic CUSUM optimal performance in detecting a shift of a given magnitude, however, are not optimal for detecting the same shift with the CUSUM of Q statistics, because the derivation of these design constants does not account for the distribution of the Q statistics following a shift. The extent of the suboptimality is greater the earlier the shift occurs. In this article, improved design constants for the CUSUM of Q statistics are obtained by using a search procedure in conjunction with Monte Carlo simulation.