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A note on coverage error of bootstrap confidence intervals for quantiles. (English) Zbl 0799.62044

Summary: An interesting recent paper by M. Falk and E. Kaufmann [Ann. Stat. 19, No. 1, 485-495 (1991; Zbl 0725.62043)] notes, with an element 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 contrast, the error would be \(O(n^{-1})\) for two-sided intervals in more classical problems, such as intervals for means or variances.
We point out that the relatively poor performance in the case of quantiles is shared by a variety of related procedures. The coverage 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 estimator as part of scale estimation, can have coverage error \(O(n^{-1 + \varepsilon})\), for arbitrarily small \(\varepsilon > 0\). Smoothed and unsmoothed versions of bootstrap percentile-\(t\) are also analysed.

MSC:

62G15 Nonparametric tolerance and confidence regions
62G09 Nonparametric statistical resampling methods

Citations:

Zbl 0725.62043
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References:

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