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Convergence of bootstrap standard error estimate (one of the problems from Efron's book)

Cross Validated Asked by Mattjosh on October 26, 2020

I am trying to solve problem 2.5 from the Efron/Tibshirani book, An Introduction to the Bootstrap (page 16).

The problem asks to show that by applying the weak law of large numbers, the bootstrap estimate of standard error

$hat{se}_{boot} = left{ sum_{b=1}^{B} [s(x^{*b}) – s(cdot)]^2 / (B-1) right}^{1/2}$

approaches

$left{ sum_{i=1}^{n} (x_i – bar{x})^2 / n^2 right}^{1/2} $

when the number of bootstrap samples, $B$, goes to infinity.

$x_i$ are the elements of the original sample which is of size $n$, $x^{*b}$ is the b-th bootstrap sample, $s(cdot) = sum_{b=1}^B s(x^{*b})/B$, and $s(x)$ is the mean $bar{x}$.

I can see empirically that this seems to be true, though I wonder how the proof is performed analytically.

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