An Elementary Approach to Weak Convergence for Quantile Processes, with Applications to Censored Survival Data

Let ξ be a continuously differentiable function with positive derivative, and let ξ n be a sequence of right-continuous increasing processes. We show that if n 1/2(ξ n − ξ) W, where W is continuous, then n 1/2(ξ−1 n − ξ−1) − W(ξ−1)/ξ′(ξ−1). This result is applied to classical processes such as the empirical distribution function, the Kaplan-Meier estimator, and some other situations. We also prove an analogous result for the bootstrapped version of n 1/2(ξ n − ξ) and show how this allows one to obtain confidence bands for the quantile function ξ−1, based on the bootstrap. Several examples are given.