Generalized Rank Regression

Rank regression offers robustness to outliers and heavy-tailed response distributions, invariance to monotonic transformations, and improved efficiency under non-Gaussian errors, making it a versatile tool for analyzing complex data. This paper introduces Generalized Rank Regression (GRR), an extension of classical rank-based methods that accommodates non-monotonic score functions. While aimed at enhancing the statistical efficiency of robust estimators, this generalization results in a potentially non-convex and non-smooth objective function, presenting challenges for both theoretical analysis and algorithmic implementation. We derive a non-asymptotic Bahadur representation of the proposed estimator and establish its asymptotic normality under mild conditions. To address the optimization challenges, we propose a new two-stage sub-gradient descent algorithm that enables efficient computation of GRR estimators with desirable statistical properties. Furthermore, we develop a multiplier bootstrap procedure for conducting statistical inference. A close connection between GRR and variants of quantile regression is uncovered, which demonstrates that GRR and composite quantile regression share asymptotically equivalent variances. The advantages of GRR are illustrated through extensive simulation studies and a real data application.