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:

$$SD = \sqrt{\frac{1}{1000-1}\sum_{r=1}^{1000}(\hat{\alpha}_{r}-\bar{\alpha})^{2}} = 0.083$$