Bootstrap methods: another look at the jackknife
Open accessAbstract
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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Slow eye movements and EEG power spectra during wake-sleep transition
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- Authors
- Efron, B.
- Affiliations
- Stanford University (US)
- Corresponding author
- Efron, B.
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical 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
- bronze
- doi.org
- 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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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