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Jackknife Estimation

The Jackknife estimation method is first used by Quenouille (1956) (Link to the paper: http://links.jstor.org/sici?sici=0006-3444%28195612%2943%3A3%2F4%3C353%3ANOBIE%3E2.0.CO%3B2-4) and Jones (1956) (http://links.jstor.org/sici?sici=0162-1459%28195603%2951%3A273%3C54%3AITPOAS%3E2.0.CO%3B2-O).

A simple case

Suppose we have a sample $x=(x_1,x_2,...,x_n)$ and an estimator $\hat{\theta}=s(x)$ . The jackknife
uses the samples that leave out one observation at a time:
$x_{(i)}=(x_1,x_2,...,x_{i-1}, x_{i+1},...,x_n)$ .

which is called jackknife samples. The ith jackknife sample consists of the data set
with the ith observation removed. Let $\hat{\theta}_{(i)}=s(x_{(i)})$ , then the jackknife estimator of $\theta$ is

$\hat{\theta}_{jack}=\hat{\theta}$

and the jackknife standard error is

$\hat{s.e.(\theta)}_{jack}=\sqrt{\frac{n-1}{n}\sum(\hat{\theta}_{(i)}-\hat{\theta}_{(.)})^2}$

with $\hat{\theta}_{(.)}=\sum\hat{\theta}_{(i)}/n$ .

The General Case

Divide the sample of size $n$ into g groups of size m each, so $n = mg$ . (Often $m$ = 1 and $g = n$ .) Let $\hat{\theta}_{(j)}$ be the estimator for $\theta$ obtained by ignoring the $j$ th group and using the only using the other $g-1$ other groups.

The Jackknife estimator is $\hat{\theta}_{jack}=g\hat{\theta}-(g-1)\hat{\theta}_{(.)}$ , where $\hat{\theta}_{(.)}=\sum_{j=1}^g\hat{\theta}_{(j)}/g$ .

The benifit of Jackknife estimator is that The Jackknife estimator lowers the bias from order $1/n$ to $1/{n^2}$ .

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Jackknife Estimation

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