On the Conditional In-Control Performance of two Nonparametric Control Charts for Location with Unknown Mean or Median

Abstract

Statistical process monitoring deals with the problem of assessing whether a process is in statistical control or not, through the use of control charts. Many of these charts rely on the knowledge of in-control parameters. However, when they are unknown, practitioners use an in-control sample to make estimations, or as a reference sample to follow a nonparametric procedure when no distribution function can be assumed. The effect of using estimates instead of known parameters, and how to deal with the problems that arise, have been studied over different control charts and practical situations. Nevertheless, the corresponding research on nonparametric control charts is scarce. This research attempts to reduce this gap by measuring, through the use of Monte Carlo simulations, the conditional effect of a reference sample on the in-control and out-of-control performances of two nonparametric CUSUM control charts based on the Wilcoxon and sequential normal scores statistics, in terms of the average run length and its variability. A nonparametric bootstrap procedure was also developed and evaluated to assist practitioners in designing ad-hoc control limits with the desired performance. The reference sample size was found to have a significant and negative effect on the average run length over both charts; simulations showed a need for large samples for relatively adequate performance. However, in almost all scenarios, the alternative based on sequential normal scores showed a smaller practitioner-to-practitioner variation. After the control limits calibration via bootstrap, the asymptotic results of the proposed nonparametric bootstrap showed a bias in performance.