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Bootstrapping works well to access the uncertainty in the mean estimate, however I remember reading somewhere the bootstrap does not do a good job in assessing the uncertainty in quantile estimates (particularly the median).

I don't remember where I read this, and I couldn't find much with a quick Google search. Thoughts on this and any references would be greatly appreciated.

whuber
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Glen
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  • It sounds strange to me, as bootstrapping is how the sqreg (simultaneous-quantile regression) command in Stata estimates the standard errors. But this does not prove anything, I know. – boscovich Aug 28 '12 at 15:42
  • See also: Rogers, W. H. 1992. sg11: Quantile regression standard errors. Stata Technical Bulletin 9: 16–19. Reprinted in Stata Technical Bulletin Reprints, vol. 2, pp. 133–137. College Station, TX: Stata Press. --- Rogers, W. H. 1993. sg11.2: Calculation of quantile regression standard errors. Stata Technical Bulletin 13: 18–19. Reprinted in Stata Technical Bulletin Reprints, vol. 3, pp. 77–78. College Station, TX: Stata Press. – boscovich Aug 28 '12 at 15:46
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    The reference you mention might be related to (1) A Note on Bootstrapping the Sample Median, (2) Exact convergence rate of bootstrap quantile variance estimator –  Aug 28 '12 at 16:07
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    I wonder if there was a miscommunication. It is well understood that the bootstrap works better in the middle of a distribution than at the tails. Thus, eg, bootstrapping the median would be the most robust quantile, whereas bootstrapping the min or max necessarily fails. You may find @cardinal's answer here to be of interest. – gung - Reinstate Monica Aug 28 '12 at 16:20
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    @Procrastinator Thank you for the two very relevant references that you cite. My book that I cite in my answer is loaded with references to bootstrap articles and both the references that you cite are listed in the book. – Michael R. Chernick Aug 28 '12 at 16:23

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The median can be bootstrapped and estimation of the median is a good application of the bootstrap. Staudte and Sheather (1990, pp.83-850 described here derive the exact calculation of the bootstrap estimate of the standard error of the estimate of the median that was originally derived in a paper by Maritz and Jarrett in 1978. Details of this can be found on pages 48-50 of my book on the bootstrap here on amazon.com.

Michael R. Chernick
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    (+1) Although a bit of attention has to be paid to the convergence of the bootstrap variance estimator as mentioned in references I posted. From (1) "The natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example to be false unless a tail condition is imposed on $F$". –  Aug 28 '12 at 16:13
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    @Procrastinator Yes it is good that you pointed out the mild restriction for consistency. Basically, it requires a moment alpha>0 to exist. – Michael R. Chernick Aug 28 '12 at 17:55
  • (+1) @Michael, I was expecting to see an answer from you in this question. –  Aug 28 '12 at 17:58
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    @Procrastinator Yes my eyes light up when I see the term bootstrap in the question. – Michael R. Chernick Aug 28 '12 at 18:05