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

Jackknife estimators allow to correct for the bias and the error 
of the bias. The method was introduced in the 1950s in papers by 
Quenouille and Tukey. 
% We focus on a simple Fortran implementation~\cite{Be92}.  
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. \vskip -9pt

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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% \caption{\small The largest cluster surface for the 10-state (left) 
% and 2-state (right) Potts models on various lattice sizes as indicated 
% in the figure (the extensions are the value of $n_{\beta}$ or e for 
% equilibrium). }
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