Jackknife method

The jackknife method ( English 'pocket knife') is a method of resampling in statistics . Jackknife is used to the random error of an estimation method and a possible distortion (engl. Bias ) to estimate. The jackknife method is a special case of the bootstrapping process . The method was first published in 1956 and 1958 by MH Quenouille and John W. Tukey . The name is intended to emphasize the general applicability of the method for statistical purposes.

method

Jackknife is often equated with delete-1 Jackknife . In this case, from the original sample in each case a value is calculated is omitted and the estimator for this reduced sample. If not just one value is omitted from the original sample, but d many, then one speaks of delete-d jackknife . By omitting total values, different reduced samples can be generated that have many values. ${\displaystyle x_{1}\dots x_{N}}$ ${\displaystyle d}$ ${\displaystyle N}$ ${\displaystyle {\binom {N}{d}}}$ ${\displaystyle N-d}$

The following describes the delete-1 jackknife method. The mean value of the reduced jackknife sample i, which results from deleting the value , is: ${\displaystyle X_{i}}$

${\displaystyle A_{(i)}:={\frac {1}{N-1}}\sum _{k\neq i}x_{k}}$

and be the mean of the original sample . Then the mean value over all Jackknife samples is given by: ${\displaystyle {\hat {A}}}$

${\displaystyle A_{(*)}:={\frac {1}{N}}\sum _{i=1}^{N}A_{(i)}}$.

The variance of the estimator can now be estimated by:

${\displaystyle \operatorname {var} (x)={\frac {N-1}{N}}\sum _{i=1}^{N}\left(A_{(i)}-{A}_{(*)}\right)^{2}}$.

The Jackknife method returns the value for the bias of the estimator : ${\displaystyle B}$

${\displaystyle B=(N-1)({A}_{(*)}-{\hat {A}})}$

literature

Joseph Lee Rodgers: The Bootstrap, the Jackknife, and the Randomization Test: A Sampling Taxonomy . Multivariate Behavioral Research, 34, No. 4 pp. 441ff (1999)

Individual evidence

1. ↑ MH Quenouille: Notes on bias in estimation. Biometrika, 43, p. 353ff (1956)
2. ^ JW Tukey: Bias and confidence in not quite large samples. Annls. Math. Stat. 29, p. 614 (1958)
3. ↑ Bradley Efron, Charles Stein: The Jackknife Estimate of Variance. The Annals of Statistics, 9 (3), pp. 586-596 (1981)