The jackknife estimator 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 ${\displaystyle x}$ , then for a given set of i.i.d. observations ${\displaystyle x_{1},...,x_{n}}$  the natural estimator is the sample mean:

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

where the last sum used another way to indicate that the index ${\displaystyle i}$  runs over the set ${\displaystyle [n]=\{1,\ldots ,n\}}$ .

Then we proceed as follows: For each ${\displaystyle i\in [n]}$  we compute the mean ${\displaystyle {\bar {x}}_{(i)}}$  of the jackknife subsample consisting of all but the ${\displaystyle i}$ -th data point, and this is called the ${\displaystyle i}$ -th jackknife replicate:

${\displaystyle {\bar {x}}_{(i)}={\frac {1}{n-1}}\sum _{j\in [n],j\neq i}x_{j},\quad \quad i=1,\dots ,n.}$ 

It could help to think that these ${\displaystyle n}$  jackknife replicates ${\displaystyle {\bar {x}}_{(1)},\ldots ,{\bar {x}}_{(n)}}$  give us an approximation of the distribution of the sample mean ${\displaystyle {\bar {x}}}$  and the larger the ${\displaystyle n}$  the better this approximation will be. Then finally to get the jackknife estimator we take the average of these ${\displaystyle n}$  jackknife replicates:

${\displaystyle {\bar {x}}_{\mathrm {jack} }={\frac {1}{n}}\sum _{i=1}^{n}{\bar {x}}_{(i)}.}$ 

One may ask about the bias and the variance of ${\displaystyle {\bar {x}}_{\mathrm {jack} }}$ . From the definition of ${\displaystyle {\bar {x}}_{\mathrm {jack} }}$  as the average of the jackknife replicates one could try to calculate explicitly, and the bias is a trivial calculation but the variance of ${\displaystyle {\bar {x}}_{\mathrm {jack} }}$  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:

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

This establishes the identity ${\displaystyle {\bar {x}}_{\mathrm {jack} }={\bar {x}}}$ . Then taking expectations we get ${\displaystyle E[{\bar {x}}_{\mathrm {jack} }]=E[{\bar {x}}]=E[x]}$ , so ${\displaystyle {\bar {x}}_{\mathrm {jack} }}$  is unbiased, while taking variance we get ${\displaystyle V[{\bar {x}}_{\mathrm {jack} }]=V[{\bar {x}}]=V[x]/n}$ . 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.

${\displaystyle {\bar {x}}_{\mathrm {jack} }}$  could be used to construct an empirical estimate of the bias of ${\displaystyle {\bar {x}}}$ , namely ${\displaystyle {\widehat {\operatorname {bias} }}({\bar {x}})_{\mathrm {jack} }=c({\bar {x}}_{\mathrm {jack} }-{\bar {x}})}$  with some suitable factor ${\displaystyle c>0}$ , although in this case we know that ${\displaystyle {\bar {x}}_{\mathrm {jack} }={\bar {x}}}$  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 ${\displaystyle {\bar {x}}}$  can be calculated from the variance of the jackknife replicates ${\displaystyle {\bar {x}}_{(i)}}$ :^[3][4]

${\displaystyle {\widehat {\operatorname {var} }}({\bar {x}})_{\mathrm {jack} }={\frac {n-1}{n}}\sum _{i=1}^{n}({\bar {x}}_{(i)}-{\bar {x}}_{\mathrm {jack} })^{2}={\frac {1}{n(n-1)}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.}$ 

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

The jackknife technique can be used to estimate (and correct) the bias of an estimator calculated over the entire sample.

Suppose ${\displaystyle \theta }$  is the target parameter of interest, which is assumed to be some functional of the distribution of ${\displaystyle x}$ . Based on a finite set of observations ${\displaystyle x_{1},...,x_{n}}$ , which is assumed to consist of i.i.d. copies of ${\displaystyle x}$ , the estimator ${\displaystyle {\hat {\theta }}}$  is constructed:

${\displaystyle {\hat {\theta }}=f_{n}(x_{1},\ldots ,x_{n}).}$ 

The value of ${\displaystyle {\hat {\theta }}}$  is sample-dependent, so this value will change from one random sample to another.

By definition, the bias of ${\displaystyle {\hat {\theta }}}$  is as follows:

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

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

We construct the jackknife replicates:

${\displaystyle {\hat {\theta }}_{(1)}=f_{n-1}(x_{2},x_{3}\ldots ,x_{n})}$ 

${\displaystyle {\hat {\theta }}_{(2)}=f_{n-1}(x_{1},x_{3},\ldots ,x_{n})}$ 

${\displaystyle \vdots }$ 

${\displaystyle {\hat {\theta }}_{(n)}=f_{n-1}(x_{1},x_{2},\ldots ,x_{n-1})}$ 

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

${\displaystyle {\hat {\theta }}_{(i)}=f_{n-1}(x_{1},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{n})\quad \quad i=1,\dots ,n.}$ 

Then we define their average:

${\displaystyle {\hat {\theta }}_{\mathrm {jack} }={\frac {1}{n}}\sum _{i=1}^{n}{\hat {\theta }}_{(i)}}$ 

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

${\displaystyle {\widehat {\text{bias}}}({\hat {\theta }})_{\mathrm {jack} }=(n-1)({\hat {\theta }}_{\mathrm {jack} }-{\hat {\theta }})}$ 

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

${\displaystyle {\hat {\theta }}_{\text{jack}}^{*}={\hat {\theta }}-{\widehat {\text{bias}}}({\hat {\theta }})_{\mathrm {jack} }=n{\hat {\theta }}-(n-1){\hat {\theta }}_{\mathrm {jack} }.}$ 

This removes the bias in the special case that the bias is ${\displaystyle O(n^{-1})}$  and reduces it to ${\displaystyle O(n^{-2})}$  in other cases.^[2]

The jackknife technique can be also used to estimate the variance of an estimator calculated over the entire sample.

• Leave-one-out error

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