Generalized Control Charts for Non-Normal Data Using g-and-k Distributions

Abstract

Statistical control charts are often used in industry to monitor processes in the interests of quality improvement. Such charts assume independence and normality of the control statistic, but these assumptions are often violated in practice. To better capture the true shape of the underlying distribution of the control statistic, we utilize the g-and-k distributions to estimate probability limits, the true ARL, and the error in confidence that arises from incorrectly assuming normality. A sensitivity assessment reveals that the extent of error in confidence associated with control chart decision-making procedures increases more rapidly as the distribution becomes more skewed or as the tails of the distribution become longer than those of the normal distribution. These methods are illustrated using both a frequentist and computational Bayesian approach to estimate the g-and-k parameters in two different practical applications. The Bayesian approach is appealing because it can account for prior knowledge in the estimation procedure and yields posterior distributions of parameters of interest such as control limits.

Keywords:

• Average run length (ARL)
• Bayesian estimation
• Control chart
• g-and-k distributions
• Non-normality
• Robustness

Mathematics Subject Classification:

• 62
• 62F35
• 62F15
• 62P30

Acknowledgments

The authors would like to thank the Editor and referee for valuable comments and suggestions to improve this article. Many thanks to Professor H. L. MacGillivray, Dr. G. Rayner, and Dr. M. L. Gatton for computational assistance and useful discussions. We also thank Leesa Wockner for assistance with generating the graphs in this article. The second author gratefully acknowledges financial support from the Australian Research Council.