Approximate maximum likelihood estimators for linear regression with design matrix uncertainty

TL;DR

This paper introduces an approximate maximum likelihood approach for linear regression problems with uncertain design matrices, effectively handling arbitrary noise and outperforming classical methods.

Contribution

It proposes a novel saddle point-based method to estimate densities and maximize an approximate likelihood under general design matrix uncertainty.

Findings

Performs favorably against classical methods

Handles arbitrary noise in the design matrix

Provides a new approach for non-Gaussian uncertainties

Abstract

In this paper we consider regression problems subject to arbitrary noise in the operator or design matrix. This characterization appropriately models many physical phenomena with uncertainty in the regressors. Although the problem has been studied extensively for ordinary/total least squares, and via models that implicitly or explicitly assume Gaussianity, less attention has been paid to improving estimation for regression problems under general uncertainty in the design matrix. To address difficulties encountered when dealing with distributions of sums of random variables, we rely on the saddle point method to estimate densities and form an approximate log-likelihood to maximize. We show that the proposed method performs favorably against other classical methods.