My statistics text has the following diagram:

$$\mathbb{V}_F(T_n) \overbrace{\approx}^{ \text{not so small} } \mathbb{V}_{\hat{F}_n}(T_n) \overbrace{\approx}^{ \text{small} } v_{\text{boot}}$$

Where $T_n$ is a statistic, $F$ is the true CDF and $\hat{F}_n$ is the empirical distribution function, $\mathbb{V}$ denotes variance, and $v_{\text{boot}}$ is the variance of the statistic that we got from bootstrap replications.

Why is one "small" and the other is "not so small"?


1 Answer 1


The second approximation is obtained by montecarlo simulation. Hence you can make it as small as you wish taking $B_{boot} \to \infty$ where $B_{boot}$ is the number of bootstrap replications. The first approximation is due to estimating $F$ by $F_n$. By Glivenko Cantelli theorem this is small if $n$ is sufficently large. The problem is that you can not control it unless you can get as many samples as you wish i.e make $n\to \infty$.

Note that there are others ways of using bootstrap that do not use sampling from the empirical distribution. In those cases the same explanation still hold if $\hat F$ is a consistent estimator of $F$.


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