Suppose that $(X_1, Y_1),\cdots, (X_n, Y_n)$ are i.i.d. bivariate random vectors and that $\xi_p(x)$ is the $p$-quantile of $Y_1$ given $X_1 = x$ for $0 < p < 1$. Estimation of $\xi_p(x)$, when it is monotone in $x$, has been studied in the literature. In the nonparametric conditional quantile estimation one uses only some smoothness assumptions. The asymptotic properties are superior in the latter case; however, monotonicity is not guaranteed. We introduce a new estimator that enjoys both of the above properties.
"An Improved Monotone Conditional Quantile Estimator." Ann. Statist. 21 (2) 924 - 942, June, 1993. https://doi.org/10.1214/aos/1176349158