Determine adaptive, computationally efficient linear model fitting without known error distribution

Determine a computationally efficient and adaptive procedure for fitting linear regression models that does not rely on prior knowledge of the error distribution, ensuring that the estimator remains valid across non-Gaussian noise settings.

Background

The paper motivates its contribution by highlighting limitations of ordinary least squares (OLS) outside Gaussian noise settings and the need for methods that perform well when the error distribution is unknown. While biased, non-linear estimators can outperform OLS in non-Gaussian settings, the authors note the lack of a clear, principled, and practical approach that is both computationally efficient and adaptive to the noise distribution.

This open question frames the development of the authors' approach—constructing data-driven convex loss functions via score matching to attain optimal asymptotic variance among convex M-estimators—by explicitly stating the broader methodological uncertainty that their work aims to address.