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Let's say you are working with a statistic (say, the mean of the population) of a skewed distribution with a long, long tail such that confidence intervals must be very skewed to achieve reasonable coverage precision for reasonably high n (<100) samples. You can't sample anymore because it costs too much.

OK, so you think you want to bootstrap.

But why?

Why not simply transform the sample using something like the Box-Cox transform (or similar)?

When would you absolutely choose one over the other or vice-versa? It's not clear to me how to strategize between the two.

In my case, I want to construct confidence intervals to make inferences about the population mean on a non-transformed scale. So I just assume I could transform, construct intervals, then reverse-transform and save myself the trouble with the bootstrap.

This obviously is not a popular choice. But why isn't it?

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  • $\begingroup$ Do you want to know the mean or variance or whatever of your distribution or of the transformed distribution? $\endgroup$
    – Dave
    Jan 11, 2023 at 6:21
  • $\begingroup$ Why not combine the two? $\endgroup$
    – Michael M
    Jan 11, 2023 at 6:23
  • $\begingroup$ Well, knowing the mean/sample variance of the transformed distribution isn't useful to me, no. So I'd want to construct confidence intervals to make inferences about the mean on a non-transformed scale from the sampling distribution. @Dave $\endgroup$ Jan 11, 2023 at 6:24
  • $\begingroup$ @MichaelM Isn't the point of transformation that you don't have to deal with the computational overhead of resampling but instead can use a pivotal method like a t-test directly? $\endgroup$ Jan 11, 2023 at 6:25
  • $\begingroup$ So then how is the transformation helpful? $\endgroup$
    – Dave
    Jan 11, 2023 at 6:25

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If you really mean that you want to transform the sampling distribution (as an earlier version of the question as well as comments indicated was the case), the main disadvantage of bootstrap is lost. In order to have a sampling distribution that you can transform, you have to create it. If you’re in the common situation where you do not know the population distribution and cannot analytically calculate the sampling distribution, then a way (and the only way I can think of) to get that sampling distribution would be to simulate it…

…by bootstrapping the original data, calculating the mean (or whatever) for each bootstrap sample, and getting a distribution that you can transform.

By bootstrapping anyway, you lose whatever computational advantage you hoped to have. Combine that with some arbitrary-ness in the transforming, and this sounds like a losing strategy.

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  • $\begingroup$ ok lol. this makes sense, feeling very dumb. so i guess we're back to considering the case of the transformation on the sample itself.. because yes the sampling distribution needs to be created. i appreciate the help. $\endgroup$ Jan 11, 2023 at 6:55
  • $\begingroup$ @Estimatetheestimators I find the “why not transform” question to be an interesting one. You might consider posting a separate question about that. // If you make progress with that thesis, I’d be interested in your thoughts as either a self-answer to this question or a new question/self-answer. (Our etiquette does not consider it arrogant or otherwise impolite to post self-answers, even immediate self-answers. In fact, I have done both.) $\endgroup$
    – Dave
    Jan 11, 2023 at 6:57
  • $\begingroup$ sg. the thesis does the box cox transform on each statistic calculation for each bootstrap replication, so not sure it will help. but it's still not clear to me why we don't just transform the sample and then reverse it -- instead of bootstrapping all the time. $\endgroup$ Jan 11, 2023 at 6:59
  • $\begingroup$ Based on just that description, that sounds dangerous, as the Box-Cox transformation need not be the same each time. Then it’s a sampling distribution of what? $\endgroup$
    – Dave
    Jan 11, 2023 at 7:00

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