The purpose of this paper is to derive confidence bands for quantile functions using a nonparametric likelihood ratio approach. The method is easy to implement and has several appealing properties. It applies to right-censored and left-truncated data, and it does not involve density estimation or even require the existence of a density. Previous approaches (e.g., bootstrap) have imposed smoothness conditions on the density. The performance of the proposed method is investigated in a Monte Carlo study, and an application to real data is given.
"Nonparametric likelihood ratio confidence bands for quantile functions from incomplete survival data." Ann. Statist. 24 (2) 628 - 640, April 1996. https://doi.org/10.1214/aos/1032894455