The random intrinsic fast initial response of one-sided CUSUM charts

Abstract

This article analyses the performance of a one-sided cumulative sum (CUSUM) chart that is initialized using a random starting point following the natural or intrinsic probability distribution of the CUSUM statistic. By definition, this probability distribution remains stable as the chart is used. The probability that the chart starts at zero according to this intrinsic distribution is always smaller than one, which confers on the chart a fast initial response feature. The article provides a fast and accurate algorithm to compute the in-control and out-of-control average run lengths and run-length probability distributions for one-sided CUSUM charts initialized using this random intrinsic fast initial response (RIFIR) scheme. The algorithm also computes the intrinsic distribution of the CUSUM statistic and random samples extracted from this distribution. Most importantly, no matter how the chart was initialized, if no level shifts and no alarms have occurred before time τ > 0, the distribution of the run length remaining after τ is provided by this algorithm very accurately, provided that τ is not too small.

Keywords:

• Average run length
• cumulative sum charts
• Gaussian quadrature
• Markov chains
• run-length distribution
• statistical process control

Acknowledgements

This research was partially supported by the Spanish DGI Grant MTM2005-00287. Alberto Luceño thanks Professor George E. P. Box for great help during his repeated visits to the University of Wisconsin-Madison, where part of the research was performed.