Quasi-nonparametric upper tolerance regions based on the bootstrap

In this paper the upper tolerance problem for random samples will be investigated. Here we will be concerned with populations that are skewed to the right and possess a finite minimum (e.g. the Exponential distribution). The method developed here takes the form of a multiplicative factor times a quantile estimate. The multiplicative factor is simple in form and is based on bootstrapped quantiles of order statistics, though no resampling is required. The quantile estimate itself could be of any desired form; for example, it could be nonparametric, and, therefore based on order statistics as well. The proposed tolerance limit has the desirable property of allowing for the possibility of exceeding the sample maximum, and therefore capturing more probability content, while avoiding, in general, use of the extreme order statistics.