Bootstrap

The bootstrap can be used to quantify the uncertainty associated with a given estimator or statistical learning method. Start with a simple data set $Z$, randomly select $n$ observations, $Z^{\ast 1}$. Resample with replacement. This procedure is repeated $B$ times for some large value of $B$, in order to produce $B$ different bootstrap data sets,$Z^{\ast 1},Z^{\ast 2},\dots ,Z^{\ast B}$, and $B$ corresponding $\alpha$ estimates, ${\hat{\alpha }}^{\ast 1},{\hat{\alpha }}^{\ast 2},\dots ,{\hat{\alpha }}^{\ast B}$. We can compute the standard error of these bootstrap estimates using the formula:

This serves as an estimate of the standard error of $\hat{\alpha }$ estimated from the original data set.

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