Bootstrap

Repeatedly sample from the dataset with replacement. The Bootstrap method can handle more complex scenarios than the Jackknife method. It also provides more flexibility in constructing confidence intervals.

Steps

1. We take a random sample of size $n$ from the original data set with replacement ($n$ is usually also the size of the original dataset).
2. From this sample, calculate the value of the estimator.

Standard Error of the Estimator

$\widehat{\text{SE}}(\hat{\alpha })=\sqrt{\frac{{\sum }_{i=1}^B({\hat{\alpha }}^{\ast i}-\bar{\hat{\alpha }}{)}^2}{B-1}}$ The Standard Error of $\hat{\alpha }$ is the square root of the sum of square differences of $\alpha$ from each bootstrap sample and the average of $\alpha$ over all bootstrap samples divided by the number of samples less one.

Bias of the Estimator

$\widehat{\text{Bias}}(\hat{\alpha })=\bar{\hat{\alpha }}-\widehat{{\alpha }_f}$ This Bias of the estimator is the average of the estimator across all bootstrap samples less the estimate from the original data set.