Definition:Jack-Knife Resampling

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Definition

Jack-knife resampling is a statistical technique mainly used for estimation of bias in a sample estimator of a population parameter.

It involves computation of the sample estimator with the omission of one observation at a time over the entire set of observations.


Examples

Arbitrary Example

Let us attempt to determine the bias in the correlation coefficient $r$ of a sample of $n$ pairs used as an estimator of the correlation coefficient $\rho$ of the population.

The procedure is to compute successively the values $r_{\paren i}$ for $i = 1, 2, \ldots, n$ for sample identical to the original but with the $i$th sample value omitted.

Let $r_{\paren .}$ be the mean of the $r_{\paren i}$ values.

Then the jack-knife estimator of bias is given by:

$B = \paren {n - 1} \paren {r_{\paren .} - r}$


In general, this is not an exact measure of bias, but it is usually a good approximation.


Standard Errors

Jack-knifing can be used to obtain estimates of standard errors.

While the jack-knife involves only $n$ samplings of the original data, each of a specified form, the bootstrapping method often uses $N$ samplings, where $N$ is considerably greater than $n$.


Motivation

Jack-knife resampling is particularly useful when there is no analytic theory to estimate bias.

This happens, for example, when the correlation coefficient $r$ of the sample is used as an estimator of the correlation coefficient $\rho$ of the population, for other than a bivariate normal distribution.

Computation is at this stage equivalent to that for leave-one-out cross-validation.

The computations are computer-intensive, and has largely been superseded by the technique of bootstrapping.


Also see

  • Results about jack-knife resampling can be found here.


Historical Note

The technique of jack-knife resampling was pioneered by Maurice Henry Quenouille in $1949$.


Sources