AN ELEMENTARY APPROACH TO WEAK-CONVERGENCE FOR QUANTILE PROCESSES, WITH APPLICATIONS TO CENSORED SURVIVAL-DATA

Let xi be a continuously differentiable function with positive derivative, and let xi(n) be a sequence of right-continuous increasing processes. We show that if n1/2(xi(n) - xi) -->d W, where W is continuous, then n1/2(xi(n)-1 - xi-1) -->d W(xi-1)/xi(xi-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 n1/2(xi(n) - xi) and show how this allows one to obtain confidence bands for the quantile function xi-1, based on the bootstrap. Several examples are given.