5.14 Bootstrap Standard Error
The bootstrap standard error functions as an estimate of the standard error of \(\hat{\alpha}\) estimated from the original data set.
The equation below gives the standard error for \(\hat{\alpha}\):
\[SE_{B(\hat{\alpha})} = \sqrt{\frac{1}{B-1}\sum_{r=1}^{B}\left(\hat{\alpha}^{*r} - \frac{1}{B}\sum_{r'=1}^{B}\hat{\alpha}^{*r'}\right)^2}\]
where \(B\) is the number of bootstrap samples, \(*\) indicates it’s a bootstrap estimate of \(\hat{\alpha}\) and \(\frac{1}{B}\sum_{r'=1}^{B}\hat{\alpha}^{*r'}\) is the mean of \(\hat{\alpha}^{*r'}\)
This was = 0.087 in our example
Compare to the estimate we obtained using 1,000 simulated data sets from the true population:
\[SD = \sqrt{\frac{1}{1000-1}\sum_{r=1}^{1000}(\hat{\alpha}_{r}-\bar{\alpha})^{2}} = 0.083\]