  Quantile regression (QR) is becoming increasingly popular due to its
relevance in many scientific investigations. There is a great amount of work
about linear and nonlinear QR models. Specifically, nonparametric estimation of
the conditional quantiles received particular attention, due to its model
flexibility. However, nonparametric QR techniques are limited in the number of
covariates. Dimension reduction offers a solution to this problem by
considering low-dimensional smoothing without specifying any parametric or
nonparametric regression relation. Existing dimension reduction techniques
focus on the entire conditional distribution. We, on the other hand, turn our
attention to dimension reduction techniques for conditional quantiles and
introduce a new method for reducing the dimension of the predictor X. The
novelty of this paper is threefold. We start by considering a single index
quantile regression model, which assumes that the conditional quantile depends
on X through a single linear combination of the predictors, then extend to a
multi index quantile regression model, and finally, generalize the proposed
methodology to any statistical functional of the conditional distribution. The
performance of the methodology is demonstrated through simulation examples and
a real data application. Our results suggest that this method has a good finite
sample performance and often outperforms existing methods.
