4.4.2 Bootstrap estimate of the variance

For each bootstrap dataset $D_{(b)}$, $b=1,\dots ,B$, we can define a bootstrap replication

$$ \hat{\theta }_{(b)}=t(D_{(b)}) \qquad b=1,\dots ,B $$

that is the value of the statistic for the specific bootstrap sample.

Bootstrap replicationsFigure 4.1: Bootstrap replications of a dataset and boostrap statistic computation

The bootstrap approach computes the variance of the estimator $\hat{\boldsymbol {\theta }}$ through the variance of the set $\hat{\theta }_{(b)}$, $b=1,\dots ,B$, given by

\begin{equation} \text {Var}_{\text{bs}}[\hat{\boldsymbol {\theta }}]= \frac{\sum _{b=1}^B (\hat{\theta }_{(b)}-\hat{\theta }_{(\cdot )})^2}{(B-1)} \quad \text { where } \quad \hat{\theta }_{(\cdot )}=\frac{\sum_{b=1}^ B \hat{\theta }_{(b)}}{B} \end{equation}

It can be shown that if $\hat{\theta }=\hat{\mu }$, then for $B\rightarrow \infty $, the bootstrap estimate $\text{Var}_{\text {bs}}[\hat{\boldsymbol {\theta }}]$ converges to the variance $\text {Var}\left[\hat{\boldsymbol {\mu }} \right]$.