Resampled quantile functions for error estimation and a relationship to density estimation

Non-parametric resampling of the data can be useful for variance estimation and constructing confidence intervals for quantiles. Parzen et al. (1994, Biometrika, 81, 341-350) develop a technique based on resampling estimating equations that can avoid non-parametric functional estimates required to use traditional large sample variance formulas. We show that while their resampling based variance formulas ape "automatic", their performance may also be improved by adjusting the variance of the resampling mechanism. This is shown to directly relate to the usual problem of needing to adjust the span of a non-parametric density estimate used with the standard asymptotic formula to estimate standard errors for quantiles. In addition, we examine the density estimate that is obtained by the resampling method. (C) 1999 Elsevier Science B.V. All rights reserved.