The Annals of Statistics

Theoretical Comparison of Bootstrap Confidence Intervals

Peter Hall

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We develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared. In all, seven different bootstrap methods are examined, each being usable in both parametric and nonparametric contexts. Emphasis is on the way in which the methods cope with first- and second-order departures from normality. Percentile-$t$ and accelerated bias-correction emerge as the most promising of existing techniques. Certain other methods are shown to lead to serious errors in coverage and position of critical point. An alternative approach, based on "shortest" bootstrap confidence intervals, is developed. We also make several more technical contributions. In particular, we confirm Efron's conjecture that accelerated bias-correction is second-order correct in a variety of multivariate circumstances, and give a simple interpretation of the acceleration constant.

Article information

Ann. Statist., Volume 16, Number 3 (1988), 927-953.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62F25: Tolerance and confidence regions
Secondary: 62G15: Tolerance and confidence regions 62E20: Asymptotic distribution theory

Acceleration constant bias-correction bootstrap confidence interval coverage critical point interval length nonparametric bootstrap parametric bootstrap percentile-method quantile shortest confidence interval


Hall, Peter. Theoretical Comparison of Bootstrap Confidence Intervals. Ann. Statist. 16 (1988), no. 3, 927--953. doi:10.1214/aos/1176350933.

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