Data-driven nonparametric tolerance sets

Abstract

We develop two new nonstandard methods for obtaining nonparametric tolerance sets from a univariate simple random sample. The first method consists of taking the union of a certain number of the intervals between the order statistics from the sample. The second method, which generalises the first, consists of taking the union of a certain number of the intervals between a prespecified subset of the order statistics from the sample. For each method, the number of intervals to choose is determined by the coverage probability properties desired. Both methods allow the choice of intervals to be made arbitrarily and after seeing the data, but minimal length may be used as a choice criterion. We show how to find the exact coverage probability for sets obtained using either method, and we explore some properties of sets obtained using the two methods. We use an ecological data set and a simulation study to show that the small-sample performance of the two methods compares favourably with that of other nonparametric tolerance set methods in the literature.

Keywords:

• beta distribution
• Dirichlet distribution
• judgement
• order statistics
• tolerance intervals

Acknowledgements

The author thanks two anonymous referees and an associate editor for helpful comments that have improved the paper.