The Annals of Statistics

Iterated smoothed bootstrap confidence intervals for population quantiles

Yvonne H. S. Ho and Stephen M. S. Lee

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This paper investigates the effects of smoothed bootstrap iterations on coverage probabilities of smoothed bootstrap and bootstrap-t confidence intervals for population quantiles, and establishes the optimal kernel bandwidths at various stages of the smoothing procedures. The conventional smoothed bootstrap and bootstrap-t methods have been known to yield one-sided coverage errors of orders O(n−1/2) and o(n−2/3), respectively, for intervals based on the sample quantile of a random sample of size n. We sharpen the latter result to O(n−5/6) with proper choices of bandwidths at the bootstrapping and Studentization steps. We show further that calibration of the nominal coverage level by means of the iterated bootstrap succeeds in reducing the coverage error of the smoothed bootstrap percentile interval to the order O(n−2/3) and that of the smoothed bootstrap-t interval to O(n−58/57), provided that bandwidths are selected of appropriate orders. Simulation results confirm our asymptotic findings, suggesting that the iterated smoothed bootstrap-t method yields the most accurate coverage. On the other hand, the iterated smoothed bootstrap percentile method interval has the advantage of being shorter and more stable than the bootstrap-t intervals.

Article information

Ann. Statist., Volume 33, Number 1 (2005), 437-462.

First available in Project Euclid: 8 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions
Secondary: 62F40: Bootstrap, jackknife and other resampling methods 62G30: Order statistics; empirical distribution functions

Bandwidth bootstrap-t iterated bootstrap kernel quantile smoothed bootstrap Studentized sample quantile


Ho, Yvonne H. S.; Lee, Stephen M. S. Iterated smoothed bootstrap confidence intervals for population quantiles. Ann. Statist. 33 (2005), no. 1, 437--462. doi:10.1214/009053604000000878.

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