Jackknife resampling

<h2 id="a-simple-example-mean-estimation">A simple example: mean estimation</h2>
The jackknife <a href="/facts/Estimator/CbkjcKvN">estimator</a> of a parameter is found by systematically leaving out each observation from a dataset and calculating the parameter estimate over the remaining observations and then aggregating these calculations.
For example, if the parameter to be estimated is the population mean of random variable 
 
 
 
 x
 
 
 {\displaystyle x}
 
, then for a given set of <a href="/facts/Independent_and_identically_distributed_random_variables/othIRaWt">i.i.d.</a> observations 
 
 
 
 
 x
 
 1
 
 
 ,
 .
 .
 .
 ,
 
 x
 
 n
 
 
 
 
 {\displaystyle x_{1},...,x_{n}}
 
 the natural estimator is the sample mean:

x
              ¯
            
          
        
        =
        
          
            1
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          x
          
            i
          
        
        =
        
          
            1
            n
          
        
        
          ∑
          
            i
            ∈
            [
            n
            ]
          
        
        
          x
          
            i
          
        
        ,
      
    
    {\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {1}{n}}\sum _{i\in [n]}x_{i},}

where the last sum used another way to indicate that the index 
 
 
 
 i
 
 
 {\displaystyle i}
 
 runs over the set 
 
 
 
 [
 n
 ]
 =
 {
 1
 ,
 …
 ,
 n
 }
 
 
 {\displaystyle [n]=\{1,\ldots ,n\}}
 
.
Then we proceed as follows: For each 
 
 
 
 i
 ∈
 [
 n
 ]
 
 
 {\displaystyle i\in [n]}
 
 we compute the mean 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 (
 i
 )
 
 
 
 
 {\displaystyle {\bar {x}}_{(i)}}
 
 of the jackknife subsample consisting of all but the 
 
 
 
 i
 
 
 {\displaystyle i}
 
-th data point, and this is called the 
 
 
 
 i
 
 
 {\displaystyle i}
 
-th jackknife replicate:

x
                ¯
              
            
          
          
            (
            i
            )
          
        
        =
        
          
            1
            
              n
              −
              1
            
          
        
        
          ∑
          
            j
            ∈
            [
            n
            ]
            ,
            j
            ≠
            i
          
        
        
          x
          
            j
          
        
        ,
        
        
        i
        =
        1
        ,
        …
        ,
        n
        .
      
    
    {\displaystyle {\bar {x}}_{(i)}={\frac {1}{n-1}}\sum _{j\in [n],j\neq i}x_{j},\quad \quad i=1,\dots ,n.}

It could help to think that these 
 
 
 
 n
 
 
 {\displaystyle n}
 
 jackknife replicates 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 (
 1
 )
 
 
 ,
 …
 ,
 
 
 
 
 x
 ¯
 
 
 
 
 (
 n
 )
 
 
 
 
 {\displaystyle {\bar {x}}_{(1)},\ldots ,{\bar {x}}_{(n)}}
 
 approximate the distribution of the sample mean 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 {\displaystyle {\bar {x}}}
 
. A larger 
 
 
 
 n
 
 
 {\displaystyle n}
 
 improves the approximation. Then finally to get the jackknife estimator, the 
 
 
 
 n
 
 
 {\displaystyle n}
 
 jackknife replicates are averaged:

x
                ¯
              
            
          
          
            
              j
              a
              c
              k
            
          
        
        =
        
          
            1
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          
            
              
                x
                ¯
              
            
          
          
            (
            i
            )
          
        
        .
      
    
    {\displaystyle {\bar {x}}_{\mathrm {jack} }={\frac {1}{n}}\sum _{i=1}^{n}{\bar {x}}_{(i)}.}

One may ask about the bias and the variance of 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 
 
 {\displaystyle {\bar {x}}_{\mathrm {jack} }}
 
. From the definition of 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 
 
 {\displaystyle {\bar {x}}_{\mathrm {jack} }}
 
 as the average of the jackknife replicates one could try to calculate explicitly. The bias is a trivial calculation, but the variance of 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 
 
 {\displaystyle {\bar {x}}_{\mathrm {jack} }}
 
 is more involved since the jackknife replicates are not independent.
For the special case of the mean, one can show explicitly that the jackknife estimate equals the usual estimate:

1
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          
            
              
                x
                ¯
              
            
          
          
            (
            i
            )
          
        
        =
        
          
            
              x
              ¯
            
          
        
        .
      
    
    {\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}{\bar {x}}_{(i)}={\bar {x}}.}

This establishes the identity 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 =
 
 
 
 x
 ¯
 
 
 
 
 
 {\displaystyle {\bar {x}}_{\mathrm {jack} }={\bar {x}}}
 
. Then taking expectations we get 
 
 
 
 E
 [
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 ]
 =
 E
 [
 
 
 
 x
 ¯
 
 
 
 ]
 =
 E
 [
 x
 ]
 
 
 {\displaystyle E[{\bar {x}}_{\mathrm {jack} }]=E[{\bar {x}}]=E[x]}
 
, so 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 
 
 {\displaystyle {\bar {x}}_{\mathrm {jack} }}
 
 is unbiased, while taking variance we get 
 
 
 
 V
 [
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 ]
 =
 V
 [
 
 
 
 x
 ¯
 
 
 
 ]
 =
 V
 [
 x
 ]
 
 /
 
 n
 
 
 {\displaystyle V[{\bar {x}}_{\mathrm {jack} }]=V[{\bar {x}}]=V[x]/n}
 
. However, these properties do not generally hold for parameters other than the mean.
This simple example for the case of mean estimation is just to illustrate the construction of a jackknife estimator, while the real subtleties (and the usefulness) emerge for the case of estimating other parameters, such as higher moments than the mean or other functionals of the distribution.

 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 
 
 {\displaystyle {\bar {x}}_{\mathrm {jack} }}
 
 could be used to construct an empirical estimate of the bias of 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 {\displaystyle {\bar {x}}}
 
, namely 
 
 
 
 
 
 
 bias
 ^
 
 
 
 (
 
 
 
 x
 ¯
 
 
 
 
 )
 
 
 j
 a
 c
 k
 
 
 
 =
 c
 (
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 −
 
 
 
 x
 ¯
 
 
 
 )
 
 
 {\displaystyle {\widehat {\operatorname {bias} }}({\bar {x}})_{\mathrm {jack} }=c({\bar {x}}_{\mathrm {jack} }-{\bar {x}})}
 
 with some suitable factor 
 
 
 
 c
 >
 0
 
 
 {\displaystyle c>0}
 
, although in this case we know that 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 j
 a
 c
 k
 
 
 
 =
 
 
 
 x
 ¯
 
 
 
 
 
 {\displaystyle {\bar {x}}_{\mathrm {jack} }={\bar {x}}}
 
 so this construction does not add any meaningful knowledge, but it gives the correct estimation of the bias (which is zero).
A jackknife estimate of the variance of 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 {\displaystyle {\bar {x}}}
 
 can be calculated from the variance of the jackknife replicates 
 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 (
 i
 )
 
 
 
 
 {\displaystyle {\bar {x}}_{(i)}}
 
:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

var
              ^
            
          
        
        (
        
          
            
              x
              ¯
            
          
        
        
          )
          
            
              j
              a
              c
              k
            
          
        
        =
        
          
            
              n
              −
              1
            
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        (
        
          
            
              
                x
                ¯
              
            
          
          
            (
            i
            )
          
        
        −
        
          
            
              
                x
                ¯
              
            
          
          
            
              j
              a
              c
              k
            
          
        
        
          )
          
            2
          
        
        =
        
          
            1
            
              n
              (
              n
              −
              1
              )
            
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        (
        
          x
          
            i
          
        
        −
        
          
            
              x
              ¯
            
          
        
        
          )
          
            2
          
        
        .
      
    
    {\displaystyle {\widehat {\operatorname {var} }}({\bar {x}})_{\mathrm {jack} }={\frac {n-1}{n}}\sum _{i=1}^{n}({\bar {x}}_{(i)}-{\bar {x}}_{\mathrm {jack} })^{2}={\frac {1}{n(n-1)}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.}

The left equality defines the estimator 
 
 
 
 
 
 
 var
 ^
 
 
 
 (
 
 
 
 x
 ¯
 
 
 
 
 )
 
 
 j
 a
 c
 k
 
 
 
 
 
 {\displaystyle {\widehat {\operatorname {var} }}({\bar {x}})_{\mathrm {jack} }}
 
 and the right equality is an identity that can be verified directly. Then taking expectations we get 
 
 
 
 E
 [
 
 
 
 var
 ^
 
 
 
 (
 
 
 
 x
 ¯
 
 
 
 
 )
 
 
 j
 a
 c
 k
 
 
 
 ]
 =
 V
 [
 x
 ]
 
 /
 
 n
 =
 V
 [
 
 
 
 x
 ¯
 
 
 
 ]
 
 
 {\displaystyle E[{\widehat {\operatorname {var} }}({\bar {x}})_{\mathrm {jack} }]=V[x]/n=V[{\bar {x}}]}
 
, so this is an unbiased estimator of the variance of 
 
 
 
 
 
 
 x
 ¯
 
 
 
 
 
 {\displaystyle {\bar {x}}}
 
.

<h2 id="estimating-the-bias-of-an-estimator">Estimating the bias of an estimator</h2>
The jackknife technique can be used to estimate (and correct) the bias of an estimator calculated over the entire sample.
Suppose 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 is the target parameter of interest, which is assumed to be some functional of the distribution of 
 
 
 
 x
 
 
 {\displaystyle x}
 
. Based on a finite set of observations 
 
 
 
 
 x
 
 1
 
 
 ,
 .
 .
 .
 ,
 
 x
 
 n
 
 
 
 
 {\displaystyle x_{1},...,x_{n}}
 
, which is assumed to consist of <a href="/facts/I.i.d./othIRaWt">i.i.d.</a> copies of 
 
 
 
 x
 
 
 {\displaystyle x}
 
, the estimator 
 
 
 
 
 
 
 θ
 ^
 
 
 
 
 
 {\displaystyle {\hat {\theta }}}
 
 is constructed:

θ
              ^
            
          
        
        =
        
          f
          
            n
          
        
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        .
      
    
    {\displaystyle {\hat {\theta }}=f_{n}(x_{1},\ldots ,x_{n}).}

The value of 
 
 
 
 
 
 
 θ
 ^
 
 
 
 
 
 {\displaystyle {\hat {\theta }}}
 
 is sample-dependent, so this value will change from one random sample to another.
By definition, the bias of 
 
 
 
 
 
 
 θ
 ^
 
 
 
 
 
 {\displaystyle {\hat {\theta }}}
 
 is as follows:

bias
        
        (
        
          
            
              θ
              ^
            
          
        
        )
        =
        E
        [
        
          
            
              θ
              ^
            
          
        
        ]
        −
        θ
        .
      
    
    {\displaystyle {\text{bias}}({\hat {\theta }})=E[{\hat {\theta }}]-\theta .}

One may wish to compute several values of 
 
 
 
 
 
 
 θ
 ^
 
 
 
 
 
 {\displaystyle {\hat {\theta }}}
 
 from several samples, and average them, to calculate an empirical approximation of 
 
 
 
 E
 [
 
 
 
 θ
 ^
 
 
 
 ]
 
 
 {\displaystyle E[{\hat {\theta }}]}
 
, but this is impossible when there are no "other samples" when the entire set of available observations 
 
 
 
 
 x
 
 1
 
 
 ,
 .
 .
 .
 ,
 
 x
 
 n
 
 
 
 
 {\displaystyle x_{1},...,x_{n}}
 
 was used to calculate 
 
 
 
 
 
 
 θ
 ^
 
 
 
 
 
 {\displaystyle {\hat {\theta }}}
 
. In this kind of situation the jackknife resampling technique may be of help.
We construct the jackknife replicates:

θ
                ^
              
            
          
          
            (
            1
            )
          
        
        =
        
          f
          
            n
            −
            1
          
        
        (
        
          x
          
            2
          
        
        ,
        
          x
          
            3
          
        
        …
        ,
        
          x
          
            n
          
        
        )
      
    
    {\displaystyle {\hat {\theta }}_{(1)}=f_{n-1}(x_{2},x_{3}\ldots ,x_{n})}

θ
                ^
              
            
          
          
            (
            2
            )
          
        
        =
        
          f
          
            n
            −
            1
          
        
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            3
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
      
    
    {\displaystyle {\hat {\theta }}_{(2)}=f_{n-1}(x_{1},x_{3},\ldots ,x_{n})}

⋮
      
    
    {\displaystyle \vdots }

θ
                ^
              
            
          
          
            (
            n
            )
          
        
        =
        
          f
          
            n
            −
            1
          
        
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        …
        ,
        
          x
          
            n
            −
            1
          
        
        )
      
    
    {\displaystyle {\hat {\theta }}_{(n)}=f_{n-1}(x_{1},x_{2},\ldots ,x_{n-1})}

where each replicate is a "leave-one-out" estimate based on the jackknife subsample consisting of all but one of the data points:

θ
                ^
              
            
          
          
            (
            i
            )
          
        
        =
        
          f
          
            n
            −
            1
          
        
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            i
            −
            1
          
        
        ,
        
          x
          
            i
            +
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        
        
        i
        =
        1
        ,
        …
        ,
        n
        .
      
    
    {\displaystyle {\hat {\theta }}_{(i)}=f_{n-1}(x_{1},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{n})\quad \quad i=1,\dots ,n.}

Then we define their average:

θ
                ^
              
            
          
          
            
              j
              a
              c
              k
            
          
        
        =
        
          
            1
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          
            
              
                θ
                ^
              
            
          
          
            (
            i
            )
          
        
      
    
    {\displaystyle {\hat {\theta }}_{\mathrm {jack} }={\frac {1}{n}}\sum _{i=1}^{n}{\hat {\theta }}_{(i)}}

The jackknife estimate of the bias of 
 
 
 
 
 
 
 θ
 ^
 
 
 
 
 
 {\displaystyle {\hat {\theta }}}
 
 is given by:

bias
              ^
            
          
        
        (
        
          
            
              θ
              ^
            
          
        
        
          )
          
            
              j
              a
              c
              k
            
          
        
        =
        (
        n
        −
        1
        )
        (
        
          
            
              
                θ
                ^
              
            
          
          
            
              j
              a
              c
              k
            
          
        
        −
        
          
            
              θ
              ^
            
          
        
        )
      
    
    {\displaystyle {\widehat {\text{bias}}}({\hat {\theta }})_{\mathrm {jack} }=(n-1)({\hat {\theta }}_{\mathrm {jack} }-{\hat {\theta }})}

and the resulting bias-corrected jackknife estimate of 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 is given by:

θ
                ^
              
            
          
          
            jack
          
          
            ∗
          
        
        =
        
          
            
              θ
              ^
            
          
        
        −
        
          
            
              bias
              ^
            
          
        
        (
        
          
            
              θ
              ^
            
          
        
        
          )
          
            
              j
              a
              c
              k
            
          
        
        =
        n
        
          
            
              θ
              ^
            
          
        
        −
        (
        n
        −
        1
        )
        
          
            
              
                θ
                ^
              
            
          
          
            
              j
              a
              c
              k
            
          
        
        .
      
    
    {\displaystyle {\hat {\theta }}_{\text{jack}}^{*}={\hat {\theta }}-{\widehat {\text{bias}}}({\hat {\theta }})_{\mathrm {jack} }=n{\hat {\theta }}-(n-1){\hat {\theta }}_{\mathrm {jack} }.}

This removes the bias in the special case that the bias is 
 
 
 
 O
 (
 
 n
 
 −
 1
 
 
 )
 
 
 {\displaystyle O(n^{-1})}
 
 and reduces it to 
 
 
 
 O
 (
 
 n
 
 −
 2
 
 
 )
 
 
 {\displaystyle O(n^{-2})}
 
 in other cases.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="estimating-the-variance-of-an-estimator">Estimating the variance of an estimator</h2>
The jackknife technique can be also used to estimate the variance of an estimator calculated over the entire sample.

<h2 id="literature">Literature</h2>
<ul><li>Berger, Y.G. (2007). "A jackknife variance estimator for unistage stratified samples with unequal probabilities". <a href="/facts/Biometrika/hg97ERcX">Biometrika</a>. 94 (4): 953–964. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fbiomet%2Fasm072">10.1093/biomet/asm072</a>.</li>
<li>Berger, Y.G.; Rao, J.N.K. (2006). <a href="https://doi.org/10.1111%2Fj.1467-9868.2006.00555.x">"Adjusted jackknife for imputation under unequal probability sampling without replacement"</a>. <a href="/facts/Journal_of_the_Royal_Statistical_Society,_Series_B/14Twyyg1">Journal of the Royal Statistical Society, Series B</a>. 68 (3): 531–547. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1111%2Fj.1467-9868.2006.00555.x">10.1111/j.1467-9868.2006.00555.x</a>.</li>
<li>Berger, Y.G.; Skinner, C.J. (2005). "A jackknife variance estimator for unequal probability sampling". <a href="/facts/Journal_of_the_Royal_Statistical_Society,_Series_B/14Twyyg1">Journal of the Royal Statistical Society, Series B</a>. 67 (1): 79–89. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1111%2Fj.1467-9868.2005.00489.x">10.1111/j.1467-9868.2005.00489.x</a>.</li>
<li>Jiang, J.; Lahiri, P.; Wan, S-M. (2002). <a href="https://doi.org/10.1214%2Faos%2F1043351257">"A unified jackknife theory for empirical best prediction with M-estimation"</a>. <a href="/facts/The_Annals_of_Statistics/frNL5W9E">The Annals of Statistics</a>. 30 (6): 1782–810. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faos%2F1043351257">10.1214/aos/1043351257</a>.</li>
<li>Jones, H.L. (1974). "Jackknife estimation of functions of stratum means". <a href="/facts/Biometrika/hg97ERcX">Biometrika</a>. 61 (2): 343–348. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2334363">10.2307/2334363</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2334363">2334363</a>.</li>
<li>Kish, L.; Frankel, M.R. (1974). "Inference from complex samples". <a href="/facts/Journal_of_the_Royal_Statistical_Society,_Series_B/14Twyyg1">Journal of the Royal Statistical Society, Series B</a>. 36 (1): 1–37.</li>
<li>Krewski, D.; Rao, J.N.K. (1981). <a href="https://doi.org/10.1214%2Faos%2F1176345580">"Inference from stratified samples: properties of the linearization, jackknife and balanced repeated replication methods"</a>. <a href="/facts/The_Annals_of_Statistics/frNL5W9E">The Annals of Statistics</a>. 9 (5): 1010–1019. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faos%2F1176345580">10.1214/aos/1176345580</a>.</li>
<li>Quenouille, M.H. (1956). "Notes on bias in estimation". <a href="/facts/Biometrika/hg97ERcX">Biometrika</a>. 43 (3–4): 353–360. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fbiomet%2F43.3-4.353">10.1093/biomet/43.3-4.353</a>.</li>
<li>Rao, J.N.K.; Shao, J. (1992). "Jackknife variance estimation with survey data under hot deck imputation". <a href="/facts/Biometrika/hg97ERcX">Biometrika</a>. 79 (4): 811–822. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fbiomet%2F79.4.811">10.1093/biomet/79.4.811</a>.</li>
<li>Rao, J.N.K.; Wu, C.F.J.; Yue, K. (1992). "Some recent work on resampling methods for complex surveys". <a href="/facts/Survey_Methodology/w4Gtxb1I">Survey Methodology</a>. 18 (2): 209–217.</li>
<li>Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer-Verlag, Inc.</li>
<li><a href="/facts/John_Wilder_Tukey/2a5FbPWb">Tukey, J.W.</a> (1958). "Bias and confidence in not-quite large samples (abstract)". <a href="/facts/The_Annals_of_Mathematical_Statistics/R8H9bM4d">The Annals of Mathematical Statistics</a>. 29 (2): 614.</li>
<li><a href="/facts/C.F._Jeff_Wu/LnJcJxGg">Wu, C.F.J.</a> (1986). <a href="https://doi.org/10.1214%2Faos%2F1176350142">"Jackknife, Bootstrap and other resampling methods in regression analysis"</a>. <a href="/facts/The_Annals_of_Statistics/frNL5W9E">The Annals of Statistics</a>. 14 (4): 1261–1295. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faos%2F1176350142">10.1214/aos/1176350142</a>.</li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Cameron, Adrian; Trivedi, Pravin K. (2005). Microeconometrics : methods and applications. Cambridge New York: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780521848053.</li>
<li><a href="/facts/Bradley_Efron/D0f95TyH">Efron, Bradley</a>; <a href="/facts/Charles_Stein_(statistician)/zoz8RXWn">Stein, Charles</a> (May 1981). <a href="https://doi.org/10.1214%2Faos%2F1176345462">"The Jackknife Estimate of Variance"</a>. The Annals of Statistics. 9 (3): 586–596. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faos%2F1176345462">10.1214/aos/1176345462</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2240822">2240822</a>.</li>
<li>Efron, Bradley (1982). The jackknife, the bootstrap, and other resampling plans. Philadelphia, PA: Society for Industrial and Applied Mathematics. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781611970319.</li>
<li>Quenouille, Maurice H. (September 1949). <a href="https://doi.org/10.1214%2Faoms%2F1177729989">"Problems in Plane Sampling"</a>. The Annals of Mathematical Statistics. 20 (3): 355–375. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faoms%2F1177729989">10.1214/aoms/1177729989</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2236533">2236533</a>.</li>
<li>Quenouille, Maurice H. (1956). "Notes on Bias in Estimation". Biometrika. 43 (3–4): 353–360. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1093%2Fbiomet%2F43.3-4.353">10.1093/biomet/43.3-4.353</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2332914">2332914</a>.</li>
<li><a href="/facts/John_Tukey/2a5FbPWb">Tukey, John W.</a> (1958). <a href="https://doi.org/10.1214%2Faoms%2F1177706647">"Bias and confidence in not quite large samples (abstract)"</a>. The Annals of Mathematical Statistics. 29 (2): 614. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faoms%2F1177706647">10.1214/aoms/1177706647</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Efron 1982, p. 2. - Efron, Bradley (1982). The jackknife, the bootstrap, and other resampling plans. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 9781611970319. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Cameron & Trivedi 2005, p. 375. - Cameron, Adrian; Trivedi, Pravin K. (2005). Microeconometrics : methods and applications. Cambridge New York: Cambridge University Press. ISBN 9780521848053. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Cameron & Trivedi 2005, p. 375. - Cameron, Adrian; Trivedi, Pravin K. (2005). Microeconometrics : methods and applications. Cambridge New York: Cambridge University Press. ISBN 9780521848053. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Efron 1982, p. 14. - Efron, Bradley (1982). The jackknife, the bootstrap, and other resampling plans. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 9781611970319. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">McIntosh, Avery I. "The Jackknife Estimation Method" (PDF). Boston University. Avery I. McIntosh. Archived from the original (PDF) on 2016-05-14. Retrieved 2016-04-30.: p. 3. <a href="https://web.archive.org/web/20160514022307/http://people.bu.edu/aimcinto/jackknife.pdf" target="_blank">https://web.archive.org/web/20160514022307/http://people.bu.edu/aimcinto/jackknife.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Cameron & Trivedi 2005, p. 375. - Cameron, Adrian; Trivedi, Pravin K. (2005). Microeconometrics : methods and applications. Cambridge New York: Cambridge University Press. ISBN 9780521848053. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Jackknife resampling open-in-new

Jackknife resampling