Exact Nonparametric Bootstrap Confidence Bands for the Quantile Function Given Censored Data

Abstract

In this note we develop an exact bootstrap algorithm for generating confidence bands for a single quantile function or joint confidence bands for k independent quantile functions given censored data.The method utilizes the classic product-limit estimator [Kaplan, E. L., Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Am. Statist. Assoc. 52:457–481] in conjunction with what is now referred to as Steck's determinant [Steck, G. P. (1971). Rectangle probabilities for uniform order statistics and the probability that the empirical distribution function lies between two distribution functions. Ann. Math. Statist. 42:1–11]. The method is termed an exact bootstrap method in the sense that no resampling is required. Unlike other bootstrap methods based on censored data the exact bootstrap method presented here is consistent.

Key Words:

• Product-limit estimator
• Quantile function
• Survival

Mathematics Subject Classification:

• 62N01
• 62N02

Acknowledgments

We wish to thank the referee for his helpful comments and time committment. This work was supported in part by a NYSTAR Faculty Development Grant.