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\date{\color{blue} Singapore MCMC Meeting, March 2004}
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\foilhead{\r The Jackknife Approach} \b \vskip -24pt

Jackknife estimators allow to correct for a bias and its statistical
error. The method was introduced in the 1950s in papers by Quenouille 
and Tukey.  The jackknife method is {\bf recommended as the standard} 
for error bar calculations. In unbiased situations the jackknife and 
the usual error bars agree. Otherwise the jackknife estimates are 
improvements, so that one cannot loose. In particular, the jackknife
method solved the question of error propagation elegantly and with
little efforts involved. 

The unbiased estimator of the expectation value $\s \widehat{x}$ is
$$\s {\overline x} = {1\over N} \sum_{i=1}^N x_i $$
Normally bias problems occur when one estimates a non-linear 
function of $\s {\widehat x}$: 
\begin{equation}  \s
 {\widehat f} = f({\widehat x})\ . 
\end{equation}
Typically, the bias is of order $\s 1/N$:
\begin{equation} \label{f_bias} \s
 {\rm bias}\ ({\overline f})\ =\ {\widehat f} -  
 \langle {\overline f}\rangle\ =\ {a_1 \over N} + {a_2 \over N^2} 
 + O({1\over N^3})\ . 
\end{equation}
Unfortunately, we lost the ability to estimate the variance 
$\s \sigma^2 ({\overline f}) = \sigma^2 (f) / N$ via the standard equation
\begin{equation} \label{f_bad_variance} \s
 s^2 ({\overline f})\ =\ {1\over N} s^2 (f)\ =\
 {1\over N\, (N-1)} \sum_{i=1}^N (f_i - {\overline f})^2\ ,  
\end{equation}
because $\s f_i=f(x_i)$ is not a valid estimator of $\s {\widehat f}$. 
Also it is in non-trivial applications almost always a bad idea to to 
use standard error propagation formulas with the aim to deduce  
$\s \triangle {\overline f}$ from $\s \triangle {\overline x}$. 
Jackknife methods are not only easier to implement, but also more 
precise and far more {\bf robust}.

The error bar problem for the estimator $\s \overline{f}$ is 
conveniently overcome by using {\bf jackknife estimators} 
$\s {\overline f^J}$, $\s f_i^J$, defined by
\begin{equation} \label{jackknife_estimators} \s
 {\overline f^J}\ =\ {1\over N} \sum_{i=1}^N f^J_i
 ~~{\rm with}~~ f_i^J\ =\ f(x_i^J) ~~{\rm and}~~ 
   x_i^J\ =\ {1\over N-1} \sum_{k\ne i} x_k\ . 
\end{equation}
The estimator for the variance $\s \sigma^2 ({\overline f^J})$ is
\begin{equation} \label{jackknife_variance} \s
 s^2_J ({\overline f^J})\ =\ {N-1 \over N} 
 \sum_{i=1}^N (f^J_i - {\overline f^J})^2\ . 
\end{equation}
Straightforward algebra shows that in the unbiased case the estimator 
of the jackknife variance~(\ref{jackknife_variance}) reduces to the 
normal variance~(\ref{f_bad_variance}). 

Notably only of order $\s N$ (not $\s N^2$) operations are needed to
construct the jackknife averages $\s x_i^J,\ i=1,\dots ,N$ from the
original data.
 
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