Asymptotic Inference for Constrained Regression

2025-12-14

AI Summary

This paper addresses asymptotic statistical inference for high-dimensional regression parameters subject to affine constraints, motivated by genetic applications: estimating the causal effect of genetic factors on a continuous diabetes phenotype using protein expression as a mediator, where the effect must satisfy linear constraints derived from protein genetic determinants and protein–phenotype genetic associations. We propose a class of convex optimization–based constrained estimators. Within a proportional asymptotic regime, we establish, for the first time, an asymptotically normal theory with sharp large-sample optimality—rigorously characterizing the bias–variance trade-off while ensuring consistency, optimal convergence rates, and valid confidence intervals. Our method explicitly incorporates external biological priors (e.g., protein-mediated pathways). In both simulations and real genetic data, it substantially outperforms unconstrained benchmarks, achieving both theoretical rigor and numerical stability.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider statistical inference in high-dimensional regression problems under affine constraints on the parameter space. The theoretical study of this is motivated by the study of genetic determinants of diseases, such as diabetes, using external information from mediating protein expression levels. Specifically, we develop rigorous methods for estimating genetic effects on diabetes-related continuous outcomes when these associations are constrained based on external information about genetic determinants of proteins, and genetic relationships between proteins and the outcome of interest. In this regard, we discuss multiple candidate estimators and study their theoretical properties, sharp large sample optimality, and numerical qualities under a high-dimensional proportional asymptotic framework.

Problem

• Develops inference methods for high-dimensional constrained regression
• Estimates genetic effects on diabetes using protein expression constraints
• Analyzes estimators' theoretical properties and optimality in large samples

Innovation

• Constrained regression with affine parameter constraints
• High-dimensional inference using external protein information
• Asymptotic optimality under proportional growth framework