# 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.