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I have a data sample on which I apply a statistic called "$\alpha$". I then use a standard bootstrap analysis on the data which results in something like this:

enter image description here

As can be seen, the bootstrap mean (green dashed line) is very much shifted with respect to the statistic applied on the original data (black solid line). This answer provided by whuber shares some light on this issue (emphasis added):

For statistics that are not linear functions of the data (...) it would be wrong simply to substitute the bootstrap mean for the statistic's value on the data: that is not how bootstrapping works. Instead, by comparing the bootstrap mean to the data statistic we obtain information about the bias of the statistic. This can be used to adjust the original statistic to remove the bias. As such, the bias-corrected estimate thereby becomes an algebraic combination of the original statistic and the bootstrap mean. For more information, look up "BCa" (bias-corrected and accelerated bootstrap) and "ABC".

My question is: how do I "adjust the original statistic to remove the bias"?

This answer shows how to estimate the bias-corrected and accelerated bootstrap interval, but there's no mention of "adjusting" the original statistic. This SAS post also discusses the BCa but again, only the interval is discussed.

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Estimating the bias of a statistic was, as I recall, one of the original motivations for developing the bootstrap. Keep in mind the bootstrap principle:

The basic idea of bootstrapping is that inference about a population from sample data (sample → population) can be modelled by resampling the sample data and performing inference about a sample from resampled data (resampled → sample). As the population is unknown, the true error in a sample statistic against its population value is unknown. In bootstrap-resamples, the 'population' is in fact the sample, and this is known; hence the quality of inference of the 'true' sample from resampled data (resampled → sample) is measurable.

On that basis, you have found that your statistic $\alpha$ calculated from the bootstrap samples has a bias of about (2.46-2.56) or -0.10 units relative to the value that you found in your original sample. Thus if you assume the above principle, your original sample had a bias of -0.10 units from the value in the full population from which it was drawn. Thus your estimate of $\alpha$ in the full population would be 2.66, 0.10 units above the value in your original sample.

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  • $\begingroup$ Thank you Ed. One thing that is still not clear to me is that you say that I need to add the bias to the estimate on the original sample, which would shift it even further away from the bootstrap mean. This is very counter-intuitive. I guess I need to read more about the bootstrap process. $\endgroup$
    – Gabriel
    Sep 20 '20 at 13:13
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    $\begingroup$ @Gabriel what you are doing is subtracting a negative bias. So that ends up the same as adding the magnitude of the bias to the value calculated on the original sample. $\endgroup$
    – EdM
    Sep 20 '20 at 13:43
  • $\begingroup$ yes, I understand that's what I should do. But it is counter-intuitive (to me) that the bias-corrected statistic on the original sample ends up even further away from the bootstrap mean. I would have naively expected that after the bias correction, the original sample would be closer to the bootstrap mean. $\endgroup$
    – Gabriel
    Sep 20 '20 at 13:47
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    $\begingroup$ @Gabriel it does take some thinking through. For me, it came together as I forced myself to get this answer in reasonable shape after an initially poor response. Go back repeatedly to the statement of the bootstrap principle quoted above. If the value from the bootstrapped samples is lower than what you got from the data sample, then the value from the data sample must be lower than the value in the entire population. $\endgroup$
    – EdM
    Sep 20 '20 at 13:58

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