Improved robust Bayes estimators of the error variance in linear models

TL;DR

This paper develops improved robust Bayes estimators for error variance in linear models with spherically symmetric, non-Gaussian errors, including scale mixtures like the multivariate-t, demonstrating their optimality and robustness.

Contribution

It introduces a new class of estimators that are both generalized Bayes and minimax for scale mixture Gaussian errors, enhancing variance estimation robustness.

Findings

Estimators outperform traditional unbiased estimators under Stein's loss.

The proposed class is both generalized Bayes and minimax.

Estimates are robust across a wide class of error distributions.

Abstract

We consider the problem of estimating the error variance in a general linear model when the error distribution is assumed to be spherically symmetric, but not necessary Gaussian. In particular we study the case of a scale mixture of Gaussians including the particularly important case of the multivariate-t distribution. Under Stein's loss, we construct a class of estimators that improve on the usual best unbiased (and best equivariant) estimator. Our class has the interesting double robustness property of being simultaneously generalized Bayes (for the same generalized prior) and minimax over the entire class of scale mixture of Gaussian distributions.