Exploiting the Quantile Optimality Ratio to Obtain Better Confidence Intervals for Quantiles

Abstract

A standard approach to confidence intervals for quantiles requires good estimates of the quantile density. The optimal bandwidth for kernel estimation of the quantile density depends on an underlying location-scale family only through the quantile optimality ratio (QOR), which is the starting point for our results. While the QOR is not distribution-free, it turns out that what is optimal for one family often works quite well for families having similar shape. This allows one to rely on a single representative QOR if one has a rough idea of the distributional shape. Another option that we explore assumes the data can be modeled by the highly flexible generalized lambda distribution (GLD), already studied by others, and we show that using the QOR for the estimated GLD can lead to more than competitive intervals. Confidence intervals for the difference between quantiles from independent populations are also considered, with an application to heart rate data.