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For doing inference of a population parameter from a sample, under which examples is better to calculate the standard error using bootstrap distribution of the mean than directly using the standard error of the sample?

Let's take the mean of income as an example:

  1. I can take a sample and calculate the income mean, and from the sample I can calculate the standard error using sample_standard_deviation/square_root(n).

  2. I can treat my sample as the "population", and apply the bootstrap, and thus create the distribution of the mean and infer the standard error.

Which are some example on using one over the other?

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  • $\begingroup$ Did you mean "standard deviation over square root sample size" in your first bullet? $\endgroup$ Jun 20 at 16:30
  • $\begingroup$ Yep. Correcting @DemetriPananos $\endgroup$
    – marz
    Jun 20 at 16:50
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    $\begingroup$ and by "doing inference", do you mean something specifically? Would you be using the standard errors in a confidence interval, for example? $\endgroup$ Jun 20 at 16:58
  • $\begingroup$ How exactly do you propose applying the bootstrap? The SE of the mean will always be directly proportional to the SD of the bootstrap distribution of the mean, anyway. $\endgroup$
    – whuber
    Jun 20 at 17:02
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    $\begingroup$ @marz There are lots of variations for bootstrapped confidence intervals, many of which just use the bootstrap samples directly rather than an estimate of the standard error. I would suggest you use either bias corrected boot strap confidence intervals or bootstrapped t intervals to start. Both are described in this book in chapter 11. $\endgroup$ Jun 20 at 18:04

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