What is the Studentized Bootstrap 🌱

Essentially it’s where you perform nested iterations of bootstrapping. Repeat the following $b$ times:

1. Sample with replacement to get a new dataset. Treat this as your original observed dataset. Calculate the statistic of interest for this dataset ${\theta }^b$
2. Perform normal bootstrapping using this sampled dataset. ${\theta }^{b,1},\dots ,{\theta }^{b,m}$
3. Collect the standard deviation (which is std error in this case) of the statistic across the nested bootstrap for this iteration, so the standard deviation of ${\theta }^{b,1},\dots ,{\theta }^{b,m}$. So we will have a standard error associated with each ${\theta }^b$.

In normal bootstrapping we aim to approximate $(\hat{\theta }-\theta )$ with $(\tilde{\theta }-\hat{\theta })$. In studentized bootstrapping we aim to approximate $(\hat{\theta }-\theta )/SE\left(\hat{\theta }\right)$ with $(\tilde{\theta }-\hat{\theta })/SE\left(\tilde{\theta }\right)$.

For more see: https://www.stat.cmu.edu/~ryantibs/advmethods/notes/bootstrap.pdf