Generalized Rank Regression

2026-05-22

AI Summary

This work proposes generalized rank regression (GRR), a novel framework that overcomes the limitations of traditional rank regression by incorporating non-monotone score functions for the first time, thereby reconciling robustness with statistical efficiency. Theoretically, the authors establish a non-asymptotic Bahadur representation and asymptotic normality for GRR estimators and demonstrate their asymptotic equivalence to composite quantile regression. Computationally, they develop an efficient two-stage subgradient descent algorithm to handle the resulting non-convex and non-smooth objective, coupled with a multiplier bootstrap procedure for valid statistical inference. Extensive simulations and empirical analyses confirm that GRR maintains robustness against outliers and heavy-tailed errors while achieving substantially higher estimation efficiency than conventional methods.

Abstract

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.

Problem

• rank regression
• non-monotonic score functions
• statistical efficiency
• non-convex optimization
• robust estimation

Innovation

• Generalized Rank Regression
• non-convex optimization
• Bahadur representation
• multiplier bootstrap
• composite quantile regression