# Jackknifing

## Jackknifing

### 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