Abstract: The quantile regression method is a valuable complement to the classical mean regression, helping to ensure robust and comprehensive data analyses in a variety of applications. We propose a novel envelope quantile regression (EQR) method that adapts a nascent technique called enveloping to improve the efficiency of the standard quantile regression. The proposed method aims to identify the material and immaterial information in a quantile regression model, and then use only the material information for estimation. By excluding the immaterial information, the EQR method has the potential to substantially reduce estimation variability. Unlike existing envelope model approaches, which rely mainly on the likelihood framework, our proposed estimator is defined through a set of nonsmooth estimating equations. We facilitate the estimation via the generalized method of moments, and derive the asymptotic normality of the proposed estimator by applying empirical process techniques. Furthermore, we establish that the EQR is asymptotically more efficient than (or at least as asymptotically efficient as) the standard quantile regression estimators, without imposing stringent conditions. Hence, our work advances the envelope model theory to general distribution-free settings. We demonstrate the effectiveness of the proposed method via Monte Carlo simulations and real data examples.

Key words and phrases: Asymptotic efficiency, envelope model, generalized method of moments, reducing subspace, sufficient dimension reduction.