Improved nonparametric tolerance intervals based on interpolated and extrapolated order statistics

^†This report is released to inform interested parties of research and to encourage discussion. The views expressed are those of the authors and not necessarily those of the US Census Bureau.

Abstract

The standard approach to construct nonparametric tolerance intervals is to use the appropriate order statistics, provided a minimum sample size requirement is met. However, it is well-known that this traditional approach is conservative with respect to the nominal level. One way to improve the coverage probabilities is to use interpolation. However, the extension to the case of two-sided tolerance intervals, as well as for the case when the minimum sample size requirement is not met, have not been studied. In this paper, an approach using linear interpolation is proposed for improving coverage probabilities for the two-sided setting. In the case when the minimum sample size requirement is not met, coverage probabilities are shown to improve by using linear extrapolation. A discussion about the effect on coverage probabilities and expected lengths when transforming the data is also presented. The applicability of this approach is demonstrated using three real data sets.

Keywords:

• Box–Cox transformation
• fractional order statistics
• one-sided tolerance limit
• tolerance package
• two-sided tolerance interval

AMS Subject Classification::

• 62G15
• 62G30

Acknowledgements

The authors thank two anonymous referees for their helpful comments that improved the quality of this paper. We would also like to thank Fritz Scholz, retired Boeing Technical Fellow and statistician, for sharing his technical reports that were germane to the discussion in our paper.

Notes

^† This report is released to inform interested parties of research and to encourage discussion. The views expressed are those of the authors and not necessarily those of the US Census Bureau.