Variance estimation for sample quantiles using the m out of n bootstrap

Abstract

We consider the problem of estimating the variance of a sample quantile calculated from a random sample of size n. The r-th-order kernel-smoothed bootstrap estimator is known to yield an impressively small relative error of order O(n-r/(2r+1)). It nevertheless requires strong smoothness conditions on the underlying density function, and has a performance very sensitive to the precise choice of the bandwidth. The unsmoothed bootstrap has a poorer relative error of order O(n-1/4), but works for less smooth density functions. We investigate a modified form of the bootstrap, known as the m out of n bootstrap, and show that it yields a relative error of order smaller than O(n-1/4) under the same smoothness conditions required by the conventional unsmoothed bootstrap on the density function, provided that the bootstrap sample size m is of an appropriate order. The estimator permits exact, simulation-free, computation and has accuracy fairly insensitive to the precise choice of m. A simulation study is reported to provide empirical comparison of the various methods. © 2005 The Institute of Statistical Mathematics.