The jackknife

We have a statistic:

\(S(x_1, x_2,...,x_n)\)

We may want to estimate moments for this statistic, but are unable to do so.

The jackknife estimator

The jackknife is an approach for getting moments for statistics.

We start by creating \(n\) statistics each leaving out one observation.

\(\bar S_i(x_1,x_2,...x_{i-1},x_{i+1},...,x_n)\)

We define:

\(\bar S=\dfrac{1}{n}\sum_i\bar S_i\)

Moments of the jackknife estimator

We want to know the variance.

\(Var \bar S=\dfrac{n-1}{n}\sum_i(\bar S_i-\bar S)^2\).

The infintesimal jackknife

The jackknife as a weighting

In the jackknife we calculate the statistic leaving one observation out.

This is the same as weighting observations and giving one a weighting of \(0\) and the others \(1\).

The infintesimal jackknife

For the infintesimal jackknife we reduce the weight not to \(0\), but by an infintesimal amount.

Variance of jackknife estimators