Definition:Jack-Knife Resampling
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The computations are computer-intensive, and has largely been superseded by the technique of bootstrapping.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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): jack-knife
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): jack-knife