Cited by 17,602Statistical Methods and InferenceAdvanced Statistical Methods and ModelsProbability and Statistical Research

Bootstrap methods: another look at the jackknife

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Efron, B.·1979·The Annals of Statistics

Abstract

We discuss the following problem: given a random sample $\mathbf{X} = (X_1, X_2, \cdots, X_n)$ from an unknown probability distribution $F$, estimate the sampling distribution of some prespecified random variable $R(\mathbf{X}, F)$, on the basis of the observed data $\mathbf{x}$. (Standard jackknife theory gives an approximate mean and variance in the case $R(\mathbf{X}, F) = \theta(\hat{F}) - \theta(F), \theta$ some parameter of interest.) A general method, called the "bootstrap," is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.

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  • Wager (2014) establishes that the infinitesimal jackknife consistently estimates the forest variance.

    “He also establishes that the infinitesimal jackknife (Efron, 1979) consistently estimates the forest variance.”

    From A random forest guided tour (2016) · cites this paper as Efron (1979)

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  1. [2][2] BARNARD, B. (1974). Conditionality, pivotals, and robust estimation. Proceedings of the Conference on Foundational Questions in Statistical Inference. Memoirs No. 1, Dept. of Theoretical Statist., Univ. of Aarhus, Denmark. -
  2. [3][3] CRAmÉR, H. (1946). Mathematical Methods in Statistics. Princeton Univ. Press. -

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Authors
Efron, B.
Affiliations
Stanford University (US)
Corresponding author
Efron, B.
Venue
The Annals of Statistics
Published
1979-01-01
Volume
7
Issue
1
Pages
1-26
ISSN
  • 0090-5364 (ISSN-L)
  • 2168-8966
OpenAlex
W2117897510
Cited by
17,602
References
14
Citations by year
2026: 865 · 2025: 599 · 2024: 592 · 2023: 654 · 2022: 679 · 2021: 887 · 2020: 890 · 2019: 895 · 2018: 848 · 2017: 720 · 2016: 780 · 2015: 791 · 2014: 838 · 2013: 717 · 2012: 702
FWCI
31.83
Type
journal article
Language
en
Open access
Locations
  • The Annals of Statistics (publishedVersion)
  • Project Euclid (Cornell University) (submittedVersion)
Topics
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
  • Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
  • Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
Keywords
bootstrapping, jackknife, linearization, linear discriminant analysis, ratio estimation, regression coefficient estimation, nonparametric estimation

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