Suppose that for each number $t$ in [0, 1] there is a distribution with distribution function $F_t(\bullet)$ which has $p$th percentile $\xi(t)$. Consider the problem of estimating $\xi(\bullet)$ under the assumption that $\xi(\bullet)$ is monotone. An estimator which is analogous to the median regression estimator considered in Cryer, Robertson, Wright and Casady (1972), is studied. Asymptotic properties including consistency and law of the iterated logarithm results are obtained under various assumptions.
"Monotone Percentile Regression." Ann. Statist. 4 (3) 532 - 541, May, 1976. https://doi.org/10.1214/aos/1176343459