  Quantile regression (QR) is becoming increasingly popular due to its
relevance in many scientific investigations. However, application of QR can
become very challenging when dealing with high-dimensional data, making it
necessary to use dimension reduction techniques. Existing dimension reduction
techniques focus on the entire conditional distribution. We turn our attention
to dimension reduction techniques for conditional quantiles and introduce a
method that serves as an intermediate step between linear and nonlinear
dimension reduction. The idea is to apply existing linear dimension reduction
techniques on the transformed predictors. The proposed estimator, which is
shown to be root-n consistent, is demonstrated through simulation examples and
real data applications. Our results suggest that this method outperforms linear
dimension reduction for conditional quantiles.
