Jackknife Resampling

Jackknife resampling is a method for estimation of the mean and higher order moments.

Given a sample $\{x_i\}$ of size $n$ for the distribution $X$, the jackknife resampling estimates the mean by leaving out each data point systematically. $n$ estimations of the mean will be obtained, with each of the estimations $x_i$

$$ \bar x_i = \frac{1}{n-1} \sum_{j\neq i} x_j. $$

The mean of the sample is

$$ \bar x = \frac{1}{n}\sum_i \bar x_i = \frac{1}{n} \sum_i \left(\frac{1}{n-1} \sum_{j\neq i} x_j\right) = \frac{1}{n}\sum_i x_i. $$

The result is consistent with other sample mean methods. Jackknife estimates the variance of the sample

$$ \operatorname{Var}(X) = \frac{1}{n(n-1)} \sum (x_i - \bar x)^2, $$

which is different from the direct estimation of the variance.

Jackknife is also used to estimate the bias of parameters.