# Transformed Central Quantile Subspace

## TL;DR

This paper introduces a novel dimension reduction method for conditional quantiles in high-dimensional data, combining linear techniques with transformations to improve performance over existing methods.

## Contribution

It proposes a new estimator that applies linear dimension reduction on transformed predictors, bridging linear and nonlinear approaches for quantile regression.

## Key findings

- The estimator is root-n consistent.
- The method outperforms traditional linear dimension reduction.
- Validated through simulations and real data applications.

## Abstract

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.