Asymptotic Inference for Constrained Regression

TL;DR

This paper develops rigorous asymptotic inference methods for high-dimensional regression models with affine constraints, motivated by genetic studies involving protein expression data.

Contribution

It introduces new theoretical tools for constrained high-dimensional regression, analyzing estimator properties and optimality under a proportional asymptotic framework.

Findings

Proposed estimators are asymptotically optimal.

Methods perform well in high-dimensional settings.

Theoretical results are supported by numerical experiments.

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.