## 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

Zbl 0725.62043
Full Text:

### References:

 [1] DOI: 10.1093/biomet/74.3.469 · Zbl 0654.62034 [2] DOI: 10.1080/03610928908830134 · Zbl 0696.62051 [3] DOI: 10.1016/0167-7152(87)90055-1 · Zbl 0626.62047 [4] Efron, The Jackknife, the Bootstrap and Other Resampling Plans (1982) · Zbl 0496.62036 [5] DOI: 10.1214/aos/1176344552 · Zbl 0406.62024 [6] DOI: 10.1080/00949659208811374 · Zbl 0775.62107 [7] Angelis, Internat. Statist. Rev 60 pp 45– (1992) [8] DOI: 10.1214/aoms/1177698342 · Zbl 0245.62043 [9] Beran, J. Roy. Statist. Soc 55 pp 643– (1993) [10] DOI: 10.1093/biomet/74.3.457 · Zbl 0663.62045 [11] Sheather, Statistical Data Analysis Based on the L pp 203– (1987) [12] DOI: 10.1214/aop/1176996132 · Zbl 0339.60017 [13] DOI: 10.1016/0167-7152(86)90021-0 · Zbl 0585.62085 [14] Hall, J. Roy. Statist. Soc 50 pp 381– (1988) [15] DOI: 10.1214/aos/1176347135 · Zbl 0672.62051 [16] DOI: 10.1080/02331889108802305 · Zbl 0809.62031 [17] DOI: 10.1214/aos/1176350933 · Zbl 0663.62046 [18] DOI: 10.1214/aop/1176991515 · Zbl 0684.62036 [19] DOI: 10.1214/aos/1176347995 · Zbl 0725.62043 [20] DOI: 10.1016/0167-7152(92)90205-J · Zbl 0761.62016 [21] Young, J. Roy. Statist. Soc 52 pp 477– (1990) [22] DOI: 10.1093/biomet/75.2.370 · Zbl 0642.62026 [23] Silverman, Density Estimation for Statistics and Data Analysis (1986) · Zbl 0617.62042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.