Non-parametric test for two sample quantiles I would like to know if there is a non-parametric test on quantiles.
EDIT: My focus is on continuous distributions (e.g. Weibull distribution).
In a general setting we define quantiles as $Q_X(p) = inf \{x \in \mathbf{R}: p \leq P(X\leq x)\}$. I would like to test $H_0: Q_X(p) = Q_Y(p)$ vs. $H_a: Q_X(p) \neq Q_Y(p)$ for some $p \in (0,1)$.

If I understand the question, one approach to compare quantiles is Mood's median test, which is a nonparametric approach that counts the observations in each group that are greater than the global median, those less-than or equal to the global median, and applying a chi-square test of association.  This approach is essentially nonparametric in that it simply counts the number of observations in each category.  It is best if the sampled population is continuous in nature; practically speaking if many observations are tied to the global median, the test can get funky.  It would be easy to modify this test to apply to any quantile.
