# Is bootstrapping a valid method to assess the uncertainty of the median estimate?

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.

• 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
• 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 – user10525 Aug 28 '12 at 16:07
• 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 Aug 28 '12 at 16:20
• @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 Chernick Aug 28 '12 at 16:23

• (+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 $F$". – user10525 Aug 28 '12 at 16:13