Exact equal-tailed β-expectation tolerance intervals for sample variances

Abstract

The sample variance of the quality characteristic shows the variation of a product or process. When the precision is a concern, a tolerance interval (TI) for sample variances is needed. In this paper, we construct β-expectation TIs for the population of sample variances, assuming normality of the data, from the frequentist and Bayesian perspectives. The accuracy level of each TI, which is the probability that the actual coverage falls into a specified range, is calculated to evaluate the performance of the TI. Besides, the determination of the minimum number of subgroups and the minimum subgroup size is discussed to achieve the desired accuracy level. When the accuracy level, the number of subgroups and subgroup size are known, the maximum variation is also calculated to measure the goodness of the proposed TI. Comparison results show that a Bayesian TI outperforms the corresponding frequentist TI. A real example is used to demonstrate the applicability and implementations of the proposed TIs.

Keywords:

• β-expectation TI
• Sample variance
• The frequentist and Bayesian perspectives
• Accuracy level

Acknowledgements

The authors thank the editor, the associate editor and the referees for their detailed comments and suggestions, which helped improve the manuscript.

Additional information

Funding

The work of Guo is sponsored by Zhejiang Provincial Natural Science Foundation of China [Grant No. LY20G020008], National Statistical Research Plan Project [Grant No. 2019LZ34] and the National Social Science Foundation of China [Grant No. 18BTJ037].