The Annals of Statistics

On the Bootstrap and Confidence Intervals

Peter Hall

Full-text: Open access

Abstract

We derive an explicit formula for the first term in an unconditional Edgeworth-type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of "Studentized" statistics. The class includes sample mean, $k$-sample mean, sample correlation coefficient, maximum likelihood estimators expressible as functions of vector means, etc. We suggest that the bootstrap is really an empiric one-term Edgeworth inversion, with the bootstrap simulations implicitly estimating the first term in an Edgeworth expansion. This view of the bootstrap is reinforced by our discussion of the iterated bootstrap, which inverts an Edgeworth expansion to arbitrary order by simulating simulations.

Article information

Source
Ann. Statist. Volume 14, Number 4 (1986), 1431-1452.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176350168

Digital Object Identifier
doi:10.1214/aos/1176350168

Mathematical Reviews number (MathSciNet)
MR868310

Zentralblatt MATH identifier
0611.62047

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G05: Estimation

Keywords
Bootstrap central limit theorem Edgeworth expansion rate of convergence Studentized statistic

Citation

Hall, Peter. On the Bootstrap and Confidence Intervals. Ann. Statist. 14 (1986), no. 4, 1431--1452. doi:10.1214/aos/1176350168. http://projecteuclid.org/euclid.aos/1176350168.


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