In-control performance of the joint Phase II − S control charts when parameters are estimated

ABSTRACT

The issue of the effects of parameter estimation on the in-control performance of control charts has motivated researchers for several decades. In this context, recently, acknowledging what has been called by some the practitioner-to-practitioner variability, a new perspective has been advocated, namely, the study of the conditional distribution of the in-control average run length (or the conditional false-alarm rate), which is more meaningful in practice. Adopting this new perspective, some authors have analyzed the conditional distribution of the false-alarm rate (or of the in-control average run length) of and of S charts separately. However, since the and S charts are not typically used separately but together or jointly in many applications, here we study the effects of parameter estimation on the performance of the two charts applied jointly (called the joint charts). For the joint charts, defining the joint false-alarm rate as the probability that at least one of the two charts ( and S) issues a false alarm, we obtain its conditional distribution, some quantiles of interest (upper prediction bounds for it) and the number of Phase I samples required to guarantee that with a high probability the conditional joint false alarm rate will not exceed a maximum tolerated value. We assume normality, consider S_p (the square root of the pooled variance) as the Phase I estimator of the process standard deviation, and consider two possibilities regarding the chart: (1) centered at and (2) centered at a specified target value. The results show (and we formally prove) that, whereas the required number of Phase I samples may be very large for the joint charts, interestingly, it lies between the corresponding numbers of samples required by the chart and by the S chart individually; so, considering the performance of the charts from the perspective of their joint use may slightly alleviate the required number of Phase I samples.

KEYWORDS:

• conditional false alarm rate
• conditional in-control average run length
• conditional and unconditional run length distribution
• false alarm rate
• in-control average run length
• joint and S charts
• parameter estimation
• Phase I
• Phase II
• Shewhart chart

About the authors

Lorena D. Loureiro holds M.Sc. and Ph.D. degrees in Industrial Engineering by PUC-Rio and works in Fundação Oswaldo Cruz (FIOCRUZ), a foundation responsible for health surveillance in Brazil. This is her second paper on the subject of the effect of parameter estimation on control charts from the conditional performance perspective.

Eugenio K. Epprecht is an Associate Professor at the Department of Industrial Engineering of PUC-Rio. His major research interest is Statistical Process Control. He has published articles in journals such as C&OR, IIE Transactions, IJPE, IJPR, JAS, The Journal of Chemometrics, JQT, QE, QREI, QTQM, among others. He has been a member of the ISBIS (International Society for Business and Industrial Statistics) council, and has organized the 2nd International Symposium on Statistical Process Control (ISSPC’2011). He is a member of ASQ.

S. Chakraborti holds a Ph.D. in Statistics by the State University of New York; he is Professor of Statistics, Robert C. and Rosa P. Morrow Faculty Excellence Fellow, Fellow of the American Statistical Association, an Elected member of the International Statistical Institute and a Fulbright Senior Scholar to South Africa. His specialty areas are Nonparametric and Robust Statistical Inference with applications in areas such as Statistical Process Control, Survival/Reliability Analysis, Econometrics, Statistical Computing, and Extreme Values. He has over 100 publications in a variety of outlets, including national and international peer-review journals and is a co-author of the book Nonparametric Statistical Inference, Fifth Edition, published by Marcel Dekker. He has been a visiting professor at a great number of universities abroad, in India, Brazil, France, South Africa, and Turkey and has won a number of teaching and research excellence awards. He has served as an Associate Editor of Communications in Statistics for over 15 years. He is a member of the American Statistical Association and the International Statistical Institute.

Felipe S. Jardim holds an B.Sc. degree in Electrical Engineering (Decision Support Systems) by PUC-Rio and an M.Sc. degree in Industrial Engineering also in PUC-Rio, where he is currently a doctoral student. He spent a research visit to the University of Alabama, USA and presented a paper at the 2016 Joint Statistical Meetings in Chicago. His current research interests are in Statistical Process Control with an emphasis on the effects of parameter estimation on control chart performance.

Acknowledgments

The authors are grateful to two anonymous reviewers for their feedback and to the Editor, whose suggestions improved the quality of the article.

Funding

This research was partly supported by the CNPq (Brazilian Council for Scientific and Technological Development) through projects numbers 308677/2015-3 (2nd author), 401523/2014-4 (3rd author), and 201172/2016-0 (4th author) as well as by CAPES (Brazilian Coordination for the Improvement of Higher Education Personnel) through a national Ph.D. scholarship for the 4th author.