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Suppose I bootstrap the distribution of the sample mean. Normally, one would use the mean of the bootstrapped distribution as point estimate of the parameter and the s.d. as its standard error. The mean of the bootstrapped distribution is asymptotically equal to the sample estimate (i.e. for a large number of iterated draws).

Now suppose the mean, or more generally, parameter, has an asymetrical bootstrap distribution, so that the sample estimate of the parameter and the mean of the bootstrap distribution are likely to be unequal for a moderate amount of iterated draws. Should I still expect both to be asympotically equivalent, so as I increase the number of draws, both will be equal? And if this is so, is it customary practice to increase the number of iterations until they are equal before I report statistics?

In my practical case, both deviate after 1000 iterated draws. So I am unsure whether I should report the sample estimate or the mean of the bootstrap distribution of the parameter.

tomka
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  • Second sentence is just wrong. Try bootstrapping the mean of 0,0,0,0,0,0,7. Sample mean is 1 but the bootstrapped distribution won't in general be symmetrical about that. – Nick Cox Dec 10 '13 at 14:41
  • True, I think I got that part wrong. Will try to ammend it. – tomka Dec 10 '13 at 14:46
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    @NickCox Deleted second sentence, amended first sentence of the second paragraph. I think now it is correct and I hope it's clear. – tomka Dec 10 '13 at 14:48
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    It is very unusual to report the mean of the bootstrap distribution as point estimate. Honestly, I can hardly imagine a situation where this makes sense. – Michael M Dec 10 '13 at 14:56
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    Related http://stats.stackexchange.com/questions/71357/why-not-report-the-mean-of-a-bootstrap-distribution – Momo Dec 10 '13 at 14:57
  • @Momo Thank you, the related discussion is very helpful. – tomka Dec 10 '13 at 15:06
  • It's strange, I thought I had learned this in my graduate statistics class. But I seem to remember it wrong. Good to know I need to take the sample estimate with bootstrapped s.e.. In the related discussion it is explained that the bootstrapped distirbution asymptotically centers arround the sample parameter (both sample size and bootstrap draws/combinations). – tomka Dec 10 '13 at 15:13
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    @Nick One of the guiding ideas of bootstrapping is estimating and *correcting* for sampling bias. When applied like this, bootstrapping definitely can provide better estimates. See chapter 1 in Peter Hall, *The Bootstrap and Edgeworth Expansion* (Springer Verlag 1992). – whuber Dec 10 '13 at 15:14
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    @tomka There are exceptions. Consider, for instance, what happens when you attempt to bootstrap the maximum (assuming, say, that the underlying distribution is Beta): it will *always* be biased low and never will "center" around the true maximum (although asymptotically the expectation of the bootstrap maximum will approach the true maximum). – whuber Dec 10 '13 at 15:16
  • The number of bootstrap samples to use is not one that gives equal results; that is necessarily elusive and in any case what does "equal" mean here? Choosing that number is, admittedly, an art as well as a science. Look at the literature in your field to see what is standard. – Nick Cox Dec 10 '13 at 15:17
  • @whuber and Nick; How do I determine then if I should use the sample estimate or the mean of the bootstrapped distribution? In practice I may not know the distribution of my statistic, which is why I use bootstrapping in the first place. – tomka Dec 10 '13 at 15:23
  • Tomka, it depends on what parameter you are bootstrapping and what distributional assumptions you have made. Bootstrapping is (partly) an art supported by extensive theoretical work (and practical experience); it is not some automatic black-box panacea. Like any other statistical procedure, there are conditions (somewhat complex) governing its use and interpretation. – whuber Dec 10 '13 at 15:29
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    @tomka: If you have a biased estimator at hand, bootstrap sometimes helps to get a better estimate of the parameter of interest. This bootstrap bias corrected version equals twice the statistic minus the mean of the bootstrap distribution. – Michael M Dec 10 '13 at 16:03

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