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It's known that one shouldn't use bootstrap to estimate minimum and maximum of the distribution which are quantiles.

I have heard the reasoning that quantiles cannot be bootstrapped because quantile is not a sufficiently smooth function of the input.

Can one give or point to a half-rigorous explanation/proof if that is or not the case?

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  • $\begingroup$ You can bootstrap the median, which is a quantile. $\endgroup$ – gung Jun 21 '16 at 17:08
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The problem is more with extreme values of distributions rather than with quantiles per se.

If the true minimum or maximum of the distribution lies beyond the limits of your data, then no amount of bootstrap re-sampling of your data will provide estimates closer to the true minimum or maximum. This answer provides a more formal description of how big this problem is, in the case of bootstrap estimation of a maximum (or minimum) order statistic from samples of a uniform distribution.

There are also problems in trying to estimate extreme quantiles, like 1% or 99%, with the bootstrap. This answer provides a good explanation. The distribution of extreme values among bootstrap samples then has more to do with the vagaries of the re-sampling than with the underlying distribution of the population of interest.

The median, a frequently used quantile, is quite amenable to bootstrap estimation. This Cross Validated page covers that issue in some detail, with several links to further useful reading that should help in considering these issues for other quantiles.

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