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As stated here: Do we use bootstrapping with population data?

"The general idea of bootstrap is that by sampling from your data you re-create the sampling process that happened when sampling your sample from the population."

So my question is:

Suppose I have a Population (P) and I take a big sample (called S), then I apply Bootstrap to that sample S to recreate my "sampling distribution". Once I have my estimator (let´s suppose the mean) of the bootstrap, my estimator would be close to the S sample estimator, ¿right? Assuming a big sample and big resamples.

But, does this boostrap estimator also corresponds to the "real" value of the population (P)? What happens if the first sample (S) I take from the population is biased?

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  • $\begingroup$ I think this answer could help:stats.stackexchange.com/questions/157639/… $\endgroup$
    – marz
    Commented Oct 8, 2020 at 2:50
  • $\begingroup$ But another question is: If I need to pick a big random sample to be unbiased from the population, why would I use this instead of making smallers "resamples" directly from the population? In other words: why use resamples from big sample (boostrap) instead of taking smaller samples directly from population? $\endgroup$
    – marz
    Commented Oct 8, 2020 at 2:55
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    $\begingroup$ With that question you've answered your own question. Don't re-sub-sample a big sample (which can be biased from the start) of the population. Bootstrap-re-subsample the entire population directly as intended $\endgroup$
    – develarist
    Commented Oct 8, 2020 at 3:55
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    $\begingroup$ It is convenient, quick, and cheap to resample from the data you already have. It may be difficult (or impossible), time-consuming, and/or expensive to obtain new samples, however small, from the population. $\endgroup$
    – whuber ♦
    Commented Oct 8, 2020 at 15:15
  • $\begingroup$ @develarist If we have the whole population, why would we do any kind of estimation? We have the population value. I don’t follow your argument. $\endgroup$
    – Dave
    Commented Oct 13, 2020 at 11:35

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No, you’re not guaranteed to nail the population value, not with bootstrap or with any other estimation method. All of the usual caveats about estimation (bias, variance...) apply to procedures based on bootstrap.

If you had some bad luck and drew a misleading sample from your population, you’re out of luck when it comes to estimation, but that’s always true.

That’s why we like random samples, as they can’t quite assure that we will get a fairly representative sample, but they make it likely.

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