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 $F$ :

$$X_1,X_2, ... ,X_n \sim F$$ (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:

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