A NOTE ON COVERAGE ERROR OF BOOTSTRAP CONFIDENCE-INTERVALS FOR QUANTILES

An interesting recent paper by Falk and Kaufmann [11] notes, with an e lement of surprise, that the percentile bootstrap applied to construct confidence intervals for quantiles produces two-sided intervals with coverage error of size n-1/2, where n denotes sample size. By way of c ontrast, the error would be O(n-1) for two-sided intervals in more cla ssical problems, such as intervals for means or variances. In the pres ent note we point out that the relatively poor performance in the case of quantiles is shared by a variety of related procedures. The covera ge accuracy of two-sided bootstrap intervals may be improved to o(n-1/ 2) by smoothing the bootstrap. We show too that a normal approximation method, not involving the bootstrap but incorporating a density estim ator as part of scale estimation, can have coverage error O(n-1+epsilo n), for arbitrarily small epsilon > 0. Smoothed and unsmoothed version s of bootstrap percentile-t are also analysed.