# Central Quantile Subspace

## TL;DR

This paper introduces a new dimension reduction method for conditional quantiles in regression models, extending to multiple indices and general functionals, with demonstrated superior performance through simulations and real data.

## Contribution

It proposes a novel dimension reduction approach for conditional quantiles, including single and multi-index models, and generalizes to any statistical functional, improving upon existing methods.

## Key findings

- Method shows good finite sample performance.
- Often outperforms existing dimension reduction techniques.
- Effective in both simulations and real data applications.

## Abstract

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.