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So, I always thought the idea of bootstrapping was that you have a sample from which you obtain an estimator for some function of the population (like the average height). And then when you bootstrap by resampling, you get draws from the distribution of the estimator (and hence the variance). I took it for granted that the mean across the bootstrapped samples would be the same as the mean from the original sample. This is definitely true for statistics such as any average across the population.

For any nonlinear function, however, such as the percentage of "rich people" in the sample, the two means will be different. So, bootstrapping is in effect telling you that your original estimator has a different mean now (which is in most cases also the mode).

Given this bias, is it still appropriate to use bootstrapping to measure variance (you obviously won't use the bootstrapped mean in place of the original estimate)?

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Rather than representing problem in the bootstrap, this feature is sometimes used to estimate the bias in your original estimator, see for example chapter 10 of Bradley Efron and Robert Tibshirani (1993) "An Introduction to the Bootstrap". Chapman & Hall/CRC.

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  • $\begingroup$ Thanks Maarten. That makes a lot of sense. Here's a follow up question, though. Say I have two functions that I calculate based on a sample. One gives me low variance, but consistent bias (say 15%) while the other gives me more variance (say 1.5 times more), but almost no bias. Also, I have a choice of going with either one and then inferring the other based on it. Which one would you recommend I estimate directly from the sample? $\endgroup$
    – ryu576
    Commented May 5, 2013 at 1:20
  • $\begingroup$ The answer to that question depends on how you value bias and precision, which in turn depends on things that only you as the researcher know. $\endgroup$
    – Maarten Buis
    Commented May 6, 2013 at 7:52
  • $\begingroup$ Fair enough, that makes sense. $\endgroup$
    – ryu576
    Commented May 6, 2013 at 21:08
  • $\begingroup$ Here are some remarks on the trade-off between variance and bias: stata.com/statalist/archive/2013-05/msg00197.html $\endgroup$
    – Maarten Buis
    Commented May 7, 2013 at 6:37

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