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.
sqreg
(simultaneous-quantile regression) command in Stata estimates the standard errors. But this does not prove anything, I know. $\endgroup$