Conditional Rank-Rank Regression via Deep Conditional Transformation Models

2026-03-07

AI Summary

This study addresses key limitations of existing conditional rank–rank regression (CRRR) methods, which struggle with nonlinear relationships, high-order interactions, and discrete ordinal outcomes, while conventional covariate expansion approaches often lack interpretable parameters. To overcome these challenges, we propose an end-to-end framework that integrates a deep conditional transformation model (DCTM) with cross-fitting, introducing a novel ω-indexed conditional rank that unifies the treatment of both continuous and discrete outcomes. The framework accommodates complex nonlinearities and high-dimensional interactions while preserving structural interpretability. We establish asymptotic theory for the continuous case and develop an exchangeable bootstrap procedure for valid inference. Simulations demonstrate substantially improved estimation accuracy, and empirical analyses reveal strong intergenerational income persistence in the United States and pronounced gender disparities in educational mobility across generations in India.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Intergenerational mobility quantifies the transmission of socio-economic outcomes from parents to children. While rank-rank regression (RRR) is standard, adding covariates directly (RRRX) often yields parameters with unclear interpretation. Conditional rank-rank regression (CRRR) resolves this by using covariate-adjusted (conditional) ranks to measure within-group mobility. We improve and extend CRRR by estimating conditional ranks with a deep conditional transformation model (DCTM) and cross-fitting, enabling end-to-end conditional distribution learning with structural constraints and strong performance under nonlinearity, high-order interactions, and discrete ordered outcomes where the distributional regression used in traditional CRRR may be cumbersome or prone to misconfiguration. We further extend CRRR to discrete outcomes via an $\omega$-indexed conditional-rank definition and study sensitivity to $\omega$. For continuous outcomes, we establish an asymptotic theory for the proposed estimators and verify the validity of exchangeable bootstrap inference. Simulations across simple/complex continuous and discrete ordered designs show clear accuracy gains in challenging settings. Finally, we apply our method to two empirical studies, revealing substantial within-group persistence in U.S. income and pronounced gender differences in educational mobility in India.

Problem

• intergenerational mobility
• conditional rank-rank regression
• covariate-adjusted ranks
• discrete ordered outcomes
• nonlinearity

Innovation

• Deep Conditional Transformation Model
• Conditional Rank-Rank Regression
• Cross-fitting
• Distributional Regression
• Intergenerational Mobility