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.

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    $\begingroup$ @NickCox Deleted second sentence, amended first sentence of the second paragraph. I think now it is correct and I hope it's clear. $\endgroup$ – tomka Dec 10 '13 at 14:48
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    $\begingroup$ 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. $\endgroup$ – Michael M Dec 10 '13 at 14:56
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    $\begingroup$ Related stats.stackexchange.com/questions/71357/… $\endgroup$ – Momo Dec 10 '13 at 14:57
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    $\begingroup$ @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). $\endgroup$ – whuber Dec 10 '13 at 15:16
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    $\begingroup$ @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. $\endgroup$ – Michael M Dec 10 '13 at 16:03

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