Bootstrap Variance Estimation

Using bootstrapping, we can approximate ${\mathbb{V}}_{{\hat{F}}_n}(T_n)$ by simulation. Here, ${\mathbb{V}}_{{\hat{F}}_n}(T_n)$ means the variance of $T_n$ if the distribution of the data is ${\hat{F}}_n$.. This is done by simulating $X_1^{\ast },\dots ,X_n^{\ast }$ from ${\hat{F}}_n$ and then compute $T_n^{\ast }=g(X_1^{\ast },\dots ,X_n^{\ast })$. This constitutes one draw from the distribution of $T_n$.

Observation

How do we simulate $X_1^{\ast },\dots ,X_n^{\ast }$ from ${\hat{F}}_n$? Notice that ${\hat{F}}_n$ puts mass $1/n$ at each data points $X_1,\dots ,X_n$. Therefore, drawing an observation from ${\hat{F}}_n$ is equivalent to drawing one point at random from the original dataset. Thus, to simulate $X_1^{\ast },\dots ,X_n^{\ast }\sim {\hat{F}}_n$ it suffices to draw $n$ observations with replacement from $X_1,\dots ,X_n$.

Bootstrap Variance Estimation

1. Draw $X_1^{\ast },\dots ,X_n^{\ast }\sim {\hat{F}}_n$.
2. Compute $T_n^{\ast }=g(X_1^{\ast },\dots ,X_n^{\ast })$.
3. Repeat steps 1 and 2, $B$ times, to get $T_{n,1}^{\ast },\dots ,T_{n,B}^{\ast }$.
4. Let