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Re-sampling methods

There is a simple motivation to use re-sampling methods. In fact let us consider a set of independent and identically distributed data samples $ n$ of an unknown probability distribution $ F$ :

$\displaystyle X_1,X_2, ... ,X_n \sim F$ (A.1)

We can compute the sample average $ \bar{x} = \sum_{i=1}^n x_i/n$ , and then we can estimate the accuracy of $ \bar{x}$ using the standard deviation:

$\displaystyle \hat{\sigma} = \sqrt{\frac{1}{n(n-1)}\sum_{i=1}^n (x_i - \bar{x})^2}$ (A.2)

The trouble with this formula is that it does not, in any obvious way, extend to estimators other than $ \bar{x}$ . For this reason a generalized version of A.2 is introduced such that it reduces to the usual standard deviation when the chosen estimator is the average.


Claudio Attaccalite 2005-11-07