An empirical semiparametric one-sided confidence bound for lower quantiles of distributions with positive support

In many industries, the reliability of a product is often determined by a quantile of a distribution of a product's characteristics meeting a specified requirement. A typical approach to address this is to assume a parametric model and compute a one-sided confidence bound on the quantile. However, this can become difficult if the sample size is too small to reliably estimate such a parametric model. Linear interpolation between order statistics is a viable nonparametric alternative if the sample size is sufficiently large. In most cases, linear extrapolation from the extreme order statistics can be used, but can result in inconsistent coverage. In this work, we perform an empirical study to generate robust weights for linear extrapolation that greatly improves the accuracy of the coverage across a feasible range of distribution families with positive support. Our method is applied to two industrial datasets.