Robust confidence intervals for the process capability index C_pk with bootstrap improvement

Abstract

The process capability index (PCI), $C_{\mathit{\text{pk}}},$ one of the widely used tools for assessing the capability of a manufacturing process, expresses the deviation of the process mean from the midpoint of the specification limits. The $C_{\mathit{\text{pk}}}$ is known to perform well under the general assumption that the experimental data are normally distributed without contamination. Under this assumption, the sample mean and sample standard deviation are used for the estimation of the PCI. However, the sample mean and sample standard deviation are quite sensitive to data contamination and this will result in underperformance of $C_{\mathit{\text{pk}}}.$ Therefore, in this article, we propose alternatives to the conventional method by replacing the sample mean and sample standard deviation with robust location and scale estimators. We also propose a method for constructing a robust PCI $C_{\mathit{\text{pk}}}$ confidence interval which lends itself to robust statistical hypothesis testing. The robust hypothesis testing methods based on this confidence interval are shown to be quite efficient when the data are normally distributed yet also outperform the conventional method when data contamination exists.

Keywords:

• bootstrap
• confidence interval
• process capability index
• robustness
• ROC

Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments and suggestions, particularly for enhancing the concluding remarks.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of Dr. Ouyang was supported by the National Natural Science Foundation of China (No. 72072089) and the Fundamental Research Funds for the Central Universities (Grant NE2023004). The work of Professor Park was supported by the National Research Foundation of Korea (NRF) grants funded by the Korean government (MSIT) (Nos. 2022R1A2C1091319 and RS-2023-00242528).

Notes on contributors

Linhan Ouyang

Linhan Ouyang is an associate professor in the College of Economics and Management at Nanjing University of Aeronautics and Astronautics, China. He holds a BEng degree in industrial engineering from Nanchang University, P.R. China, and a PhD degree in management science and engineering from Nanjing University of Science and Technology, P.R. China. His research interests are process modeling and design of experiments.

Sanku Dey

Sanku Dey is currently working as an associate professor in the Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India. He did his MSc in Statistics in the year of 1991 from Gauhati University, Guwahati, India and PhD in Statistics (reliability theory) in the year 1998 from the same university. He has published more than 270 research articles in journals of repute. He is an associate editor of American Journal of Mathematical and Management Sciences and also the member of editorial board of several journals of repute. He is a researcher and has a good number of contributions in almost all fields of Statistics viz., distribution theory, discretization of continuous distribution, reliability theory, multicomponent stress-strength reliability, survival analysis, Bayesian inference, record statistics, statistical quality control, order statistics, lifetime performance index based on classical and Bayesian approach as well as different types of censoring schemes, etc.

Chanseok Park

Chanseok Park started college as an engineering student in the Department of Mechanical Engineering at Seoul National University and obtained a BS degree. He then received his MA in Mathematics from the University of Texas at Austin and his Doctorate in Statistics from the Pennsylvania State University. He is at present a professor of Industrial Engineering at Pusan National University. He is also a Director of Applied Statistics Laboratory in the department where he leads the applied statistics group, teaches courses, and conducts various research on quality and reliability engineering, competing risks models, robust inference, solid mechanics, etc. Before joining Pusan National University, he was a faculty member of Mathematical Sciences at Clemson University, Clemson, SC, USA from 2001 to 2015.