An Improved Monotone Conditional Quantile Estimator

Suppose that (X1, Y1),⋯, (Xn, Yn) are i.i.d. bivariate random vectors and that ξp(x) is the p-quantile of Y1given X1= x for$0 < p < 1$. Estimation of ξp(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.