If we are just interested in computing confidence intervals for the population mean $\mu$ using a sample $X_1,X_2,\dots,X_n$ of $n$ iid random variables is bootstrapping redundant if $n$ is large?

I don't see any reason why a bootstrap estimate of the confidence intervals in this case would be any better than just directly applying the CLT to construct the confidence intervals based on the asymptotic normality of the sample mean?

In fact, it seems the bootstrap confidence interval may be even less accurate than confidence intervals given directly by the CLT because afaik there are a couple of levels of approximation taking place when we use bootstrapping?

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    $\begingroup$ It really depends on what is a "large sample size" to you. See the discussion in this question (talks about t-test, but closely related to CI) and this question. Technically speaking, CLT only gives a guarantee for asymptotic normality at limit, but did not say anything on the rate of convergence. This might bite hard if you are dealing with extremely skewed population distribution. $\endgroup$ – B.Liu Apr 5 at 14:16
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    $\begingroup$ I do not remember the details anymore (there was something about Edgeworth expansions - see Hansen "Econometrics" Chapter 10), but I think bootstrap converges faster (not slower) than the CLT when estimating the mean. E.g. this lecture note by David Banks from Duke says But one can show that, as n gets large, the bootstrap is never worse than the Central Limit Theorem approximation and for many parameters it can be much better. $\endgroup$ – Richard Hardy Apr 5 at 14:59
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    $\begingroup$ How do we limit the bootstrap? Can we do all possible permutations (or as many as we like) of the sample $n$ that we took? $\endgroup$ – Sextus Empiricus Apr 5 at 16:04
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    $\begingroup$ CLT uses first-order asymptotic approximation while bootstrap can be shown to effectively use a higher-order asymptotic approximation, thus it converges faster (the convergence rate is above $\sqrt{n}$). Hansen's chapter 10 contains the details. $\endgroup$ – Richard Hardy Apr 5 at 16:45
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    $\begingroup$ @Richard I find it difficult to imagine how bootstrap can converge faster. At some point the sample distribution of the mean is so close to a normal distribution that a bootstrap will just give something like $\mu \pm c \sigma/\sqrt{n}$. $\endgroup$ – Sextus Empiricus Apr 5 at 17:39

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