The Annals of Statistics

An Improvement of the Jackknife Distribution Function Estimator

James G. Booth and Peter Hall

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In a recent paper, C. F. J. Wu showed that the jackknife estimator of a distribution function has optimal convergence rate $O(n^{-1/2})$, where $n$ denotes the sample size. This rate is achieved by retaining $O(n)$ data values from the original sample during the jackknife algorithm. Wu's result is particularly important since it permits a direct comparison of jackknife and bootstrap methods for distribution estimation. In the present paper we show that a very simple, nonempirical modification of the jackknife estimator improves the convergence rate from $O(n^{-1/2})$ to $O(n^{-5/6})$, and that this rate may be achieved by retaining only $O(n^{2/3})$ data values from the original sample. Our technique consists of mixing the jackknife distribution estimator with the standard normal distribution in an appropriate proportion. The convergence rate of $O(n^{-5/6})$ makes the jackknife significantly more competitive with the bootstrap, which enjoys a convergence rate of $O(n^{-1})$ in this particular problem.

Article information

Ann. Statist., Volume 21, Number 3 (1993), 1476-1485.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62D05: Sampling theory, sample surveys

Asymptotic normality bootstrap convergence rate distribution estimation Edgeworth expansion jackknife mixture sampling without replacement


Booth, James G.; Hall, Peter. An Improvement of the Jackknife Distribution Function Estimator. Ann. Statist. 21 (1993), no. 3, 1476--1485. doi:10.1214/aos/1176349268.

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