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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?

ManUtdBloke
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    It really depends on what is a "large sample size" to you. See the discussion in [this question](https://stats.stackexchange.com/questions/69898/t-test-on-highly-skewed-data/69967) (talks about t-test, but closely related to CI) and [this question](https://stats.stackexchange.com/questions/110418/is-bootstrapping-appropriate-for-this-continuous-data). 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. – B.Liu Apr 05 '21 at 14:16
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    I do not remember the details anymore (there was something about Edgeworth expansions - see Hansen ["Econometrics"](https://www.ssc.wisc.edu/~bhansen/econometrics/Econometrics.pdf) Chapter 10), but I think bootstrap converges faster (not slower) than the CLT when estimating the mean. E.g. [this](http://www2.stat.duke.edu/~banks/111-lectures.dir/lect13.pdf) 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.* – Richard Hardy Apr 05 '21 at 14:59
  • @B.Liu Are bootstrap confidence intervals themselves not also based on the CLT? – ManUtdBloke Apr 05 '21 at 15:32
  • @RichardHardy I really can't see how the bootstrap could convergence faster when its convergence theory is also based on the CLT afaik, and there are multiple approximations happening when the bootstrap is used as shown in [this image](https://i.stack.imgur.com/M9inV.png) from an answer to [this post](https://stats.stackexchange.com/questions/26088/explaining-to-laypeople-why-bootstrapping-works) – ManUtdBloke Apr 05 '21 at 15:49
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    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? – Sextus Empiricus Apr 05 '21 at 16:04
  • @SextusEmpiricus I am considering the conventional approach to the bootstrap for which I think about 10,000 permutations are used as a general guideline – ManUtdBloke Apr 05 '21 at 16:13
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    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. – Richard Hardy Apr 05 '21 at 16:45
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    @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 $\mu \pm c \sigma/\sqrt{n}$. – Sextus Empiricus Apr 05 '21 at 17:39
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    @SextusEmpiricus, at that point the higher-order approximation will not matter much anymore, I guess. Before that might be another story. But it has been 8 years since I studied this, so I should probably just keep quiet and let someone else answer. – Richard Hardy Apr 05 '21 at 18:35
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    @ManUtdBloke There are [many flavours of bootstrap CIs](https://influentialpoints.com/Training/bootstrap_confidence_intervals-principles-properties-assumptions.htm), and if my understanding is correct, not all are strongly related to CLT? – B.Liu Apr 05 '21 at 22:26

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