SDS Seminar Series – Richard Samworth, University of Cambridge

Apr

2025

2:00 pm – 3:00 pm

In Person (view details)

Featured Speaker(s): Richard Samworth

Cost: Free

How Should We Do Linear Regression?

Description

The Spring 2025 SDS Seminar Series continues on April 18th from 2:00 p.m. to 3:00 p.m. with Dr. Richard Samworth (Professor, Department of Pure Mathematics and Mathematical Statistics and Director, Statistical Laboratory, University of Cambridge). This event is in-person in the Avaya Room (POB 2.302).

Title: How Should We Do Linear Regression?

Abstract: In the context of linear regression, we construct a data-driven convex loss function with respect to which empirical risk minimisation yields optimal asymptotic variance in the downstream estimation of the regression coefficients. Our semiparametric approach targets the best decreasing approximation of the derivative of the log-density of the noise distribution. At the population level, this fitting process is a nonparametric extension of score matching, corresponding to a log-concave projection of the noise distribution with respect to the Fisher divergence. The procedure is computationally efficient, and we prove that our procedure attains the minimal asymptotic covariance among all convex M-estimators. As an example of a non-log-concave setting, for Cauchy errors, the optimal convex loss function is Huber-like, and our procedure yields an asymptotic efficiency greater than 0.87 relative to the oracle maximum likelihood estimator of the regression coefficients that uses knowledge of this error distribution; in this sense, we obtain robustness without sacrificing much efficiency. Numerical experiments confirm the practical merits of our proposal.