Approximate two-sided tolerance interval for sample variances

Abstract

Tolerance limits for variances are useful in quality assessments when the focus is on the precision of a quality characteristic. Two-sided tolerance intervals (limits) provide insight into a process degradation as well as improvement, in terms of process variability. Sarmiento, Chakraborti, and Epprecht constructed the exact two-sided tolerance intervals for the population of sample variances, assuming normality of the data. The required tolerance factors cannot be expressed in a closed-form and their computation is complex, depending on the numerical solutions of a system of three nonlinear equations. Motivated by this, from a practical point of view, we consider a simpler, approximate tolerance interval based on the approximate tolerance interval for the gamma distribution, which uses the Wilson–Hilferty approximation. The required tolerance factors for the proposed interval are readily obtained using existing tables and software and therefore can be implemented more easily in practice. The performance of the proposed tolerance interval is compared with that of the exact interval in terms of accuracy and robustness in simulation studies. In addition, the tolerance intervals are illustrated with a dataset from a real application. A summary and some conclusions are offered. It is seen that the proposed approximate tolerance intervals are fairly accurate, reasonably robust and being much simpler to calculate, can be useful in practical applications.

Keywords:

• chi-square distribution
• gamma distribution
• normal distribution
• sample variance
• two-sided tolerance interval and limits
• Wilson–Hilferty approximation

Additional information

Funding

This work was financially supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brazil (CAPES) - Finance Code 001 for the second author, and by the CNPq (Brazilian Council for Scientific and Technological Development) through project numbers 401523/2014-4 (for the third author) and 308677/2015-3 (for the fourth author).

Notes on contributors

Subha Chakraborti

Yuhui Yao is a PhD student in Applied Statistics at the Department of Information Systems, Statistics and Management Science in the Culverhouse College of Commerce at the University of Alabama, Tuscaloosa. He received his BS in 2011 from Guangdong University of Technology, China and MS in 2016 from the University of Alabama, Tuscaloosa. His current research interests are in developing new quality monitoring methodologies and software development. He presented papers at the 2017 and 2018 Joint Statistical Meetings in Baltimore and Vancouver, respectively. He has worked for several communication companies and banks in China between 2011 and 2014 as a data analyst.

 Martin G. C. Sarmiento holds a BSc degree in Industrial Engineering from the National University of Engineering (UNI, Peru) and an MSc degree in Production Engineering from PUC-Rio, where he is currently finishing his PhD. He has been a visiting researcher at the Department of Information Systems, Statistics and Management Science at the University of Alabama in 2017. He presented papers at the 2017 and 2018 Joint Statistical Meetings in Baltimore and Vancouver, respectively. His research interests focus on Statistical Methods for Quality Control and Improvement. He is a member of a research group working on Effects of Parameters Estimation on the Performance of Control Charts. His research has been supported by CAPES (Brazilian Coordination for the Improvement of Higher Education Personnel). He has worked at Peruvian manufacturing companies in Quality Control and Quality Management Departments for some years. He is a student member of ASA.

 Subha Chakraborti is Professor of Statistics and Morrow Faculty Fellow at the University of Alabama, Tuscaloosa, Alabama, USA. He is a Fellow of the American Statistical Association and an Elected Member of the International Statistical Institute. Professor Chakraborti has contributed in a number of research areas including censored data analysis, studies on income distribution, poverty, reliability, and general statistical inference. His current research interests include applications of statistical methods, including nonparametric methods, to the area of industrial statistics and statistical process control. He is the coauthor of the highly acclaimed book, Nonparametric Statistical Inference, fifth edition (2010) published by Taylor and Francis. Professor Chakraborti has supervised over twenty Master’s and PhD students and has been cited for his contributions in mentoring and collaborative work with students and scholars from around the world.

 Eugenio K. Epprecht is an Associate Professor at the Department of Industrial Engineering of PUC-Rio and a researcher sponsored by CNPq (the Brazilian Council for Scientific and Technological Development). His major research interest is in Statistical Process Control, in which he has published over forty articles in leading journals, two book chapters, co-authored a brazilian textbook and supervised 39 Master's and PhD students. He has been a member of the ISBIS (International Society for Business and Industrial Statistics) council, and organized the 2nd International Symposium on Statistical Process Control (ISSPC'2011).