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| In [[statistics]], the '''jackknife''' is a [[resampling (statistics)|resampling]] technique especially useful for [[variance]] and [[Bias of an estimator|bias]] estimation. The Jackknife predates other common resampling methods such as the [[bootstrap (statistics)|bootstrap]]. The jackknife [[estimator]] of a parameter is found by systematically leaving out each observation from a dataset and calculating the estimate and then finding the average of these calculations. Given a sample of size ''N'' the jackknife estimate is found by aggregating the estimates of each ''N'' − 1 estimate in the sample.
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| The jackknife technique was developed in Quenouille (1949, 1956). Tukey (1958) expanded on the technique and proposed the name "jackknife" since, like a boy scout's jackknife, it is a "rough and ready" tool that can solve a variety of problems even though specific problems may be more efficiently solved with a purpose designed tool.{{sfn|Cameron|Trivedi|2005|p=375}}
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| The jackknife represents a linear approximation of the bootstrap.{{sfn|Cameron|Trivedi|2005|p=375}}
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| ==Estimation==
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| The jackknife estimate of a parameter can be found by estimating the parameter for each subsample omitting the ''i''th observation to obtain an estimate <math>\bar{\theta}_i</math>. The overall jackknife estimator is found by averaging each of these subsample estimators{{sfn|Efron|Stein|1981|p=586}}
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| :<math>\bar{\theta}_{Jack} =\frac{1}{n} \sum_{i=1}^n (\bar{\theta}_i)</math>
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| ==Variance estimation==
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| An estimate of the variance of an estimator can be calculated using the jackknife technique.
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| : <math>\operatorname{Var}(\theta) =\sigma^2 =\frac{n-1}{n} \sum_{i=1}^n (\bar{\theta}_i - \bar{\theta}_\mathrm{Jack})^2</math>
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| where <math>\bar{\theta}_i</math> is the parameter estimate based on leaving out the ''i''th observation, and <math>\bar{\theta}_\mathrm{Jack}</math> is the jackknife estimator based on all of the samples.{{sfn|Efron|1982|p=2}} | |
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| ==Bias estimation and correction==
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| The jackknife technique can be used to estimate the bias of an estimator calculated over the entire sample.
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| :<math>\bar{\theta}_\mathrm{BiasCorrected} = N \bar{\theta}-(N-1) \bar{\theta}_{Jack}</math>{{sfn|Cameron|Trivedi|2005|p=375}}
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| This reduces bias by an order of magnitude, from <math>O(N^{-1})</math> to <math>O(N^{-2})</math>.{{sfn|Cameron|Trivedi|2005|p=375}}
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| This provides an estimated correction of bias due to the estimation method. The jackknife does not correct for a biased sample.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book | last = Cameron | first = Adrian | last2=Trivedi|first2=Pravin K.|title = Microeconometrics : methods and applications | publisher = Cambridge University Press | location = Cambridge New York | year = 2005 |ref=harv| isbn = 9780521848053 }}
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| *{{cite journal|last1=Efron|first1=B.|last2=Stein|first2=C.|title=The Jackknife Estimate of Variance|journal=The Annals of Statistics|volume=9|number=3|date=May 1981|pages=586-596|url=http://www.jstor.org.proxyau.wrlc.org/stable/2240822|ref=harv}}
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| * {{cite book | last = Efron | first = Bradley | title = The jackknife, the bootstrap, and other resampling plans | publisher = Society for Industrial and Applied Mathematics| location = Philadelphia, Pa | year = 1982 |ref=harv| isbn = 9781611970319 }}
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| * {{cite journal|last=Quenouille|first=M. H.|title=Problems in Plane Sampling|journal=The Annals of Mathematical Statistics|
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| volume=20|number=3|date=September 1949|pages=355–375|jstor=2236533}}
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| * {{cite journal|last=Quenouille|first=M. H.|title=Notes on Bias in Estimation|journal=Biometrika|year=1956|pages=353–360|jstor=2332914|volume=43|number=3-4|doi=10.1093/biomet/43.3-4.353}}
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| * {{cite journal|last=Tukey|first=J.W.|title=Bias and confidence in not quite large samples|journal=The Annals of Mathematical Statistics|year=1958}}.
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| [[Category:Statistical terminology]]
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| [[Category:Computational statistics]]
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| [[Category:Data analysis]]
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| [[Category:Statistical inference]]
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| [[Category:Resampling (statistics)]]
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