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I have been reading about bootstrapping, and sampling distributions, and find it odd that people use these techniques to describe uncertainty.

As I understand it, the sampling distribution shows uncertainty in the statistic measured on a sample by resampling the sample with replacement.

So if you bootstrap your sample, what you're really getting is a statistic of your statistic, with an uncertainty bound.

But surely we only care about the original statistic because it's an estimate of the population parameter. We want to know the original statistic's uncertainty bound, because it tells us something about the population. So, why do we care about the uncertainty bound of a statistic of the original statistic?

How does that tell us something about the population parameter?

There is a lay-person answer to this question already here:

Explaining to laypeople why bootstrapping works

My question is an extension of the above. I would like to know at a deeper mathematical level how this works, preferably with empirical evidence.

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  • $\begingroup$ Bootstrapping is a way of getting a sampling distribution of your statistic, particularly when there is no analytic solution to this. $\endgroup$
    – Peter Flom
    Commented Dec 6, 2023 at 13:20
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    $\begingroup$ Why bother? Well, because it's one of the key things to find out. You want a parameter estimate, and you want some sort of sense of how good your estimate is. Sometimes, there's no analytic solution. $\endgroup$
    – Peter Flom
    Commented Dec 6, 2023 at 14:19
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    $\begingroup$ Any book on bootstrapping with Efron among the authors will give you the mathematical background. A good keyword for searching is "plug-in estimator." $\endgroup$
    – whuber ♦
    Commented Dec 6, 2023 at 14:54
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    $\begingroup$ Thank you! @whuber $\endgroup$
    – Connor
    Commented Dec 6, 2023 at 15:21
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    $\begingroup$ One of the comments I liked when I first learned about bootstrap was, “When you can’t go back to the true distribution, a representative empirical distribution is the next-best option.” $\endgroup$
    – Dave
    Commented Dec 7, 2023 at 0:21

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