Bayesian Robustness to Outliers in Linear Regression and Ratio Estimation

TL;DR

This paper extends Bayesian robustness to linear regression through the origin with variable error variance and introduces a simple heavy-tailed error model for robust estimation of means and ratios, ensuring outliers have diminishing influence.

Contribution

It generalizes Bayesian robustness results to simple linear regression with variable error variance and proposes a straightforward heavy-tailed error model for robust ratio and mean estimation.

Findings

Whole robustness achieved in simple linear regression with variable error variance.

Heavy-tailed error assumption enables robust Bayesian estimation.

Method maintains standard posterior inference procedures.

Abstract

Whole robustness is a nice property to have for statistical models. It implies that the impact of outliers gradually vanishes as they approach plus or minus infinity. So far, the Bayesian literature provides results that ensure whole robustness for the location-scale model. In this paper, we make two contributions. First, we generalise the results to attain whole robustness in simple linear regression through the origin, which is a necessary step towards results for general linear regression models. We allow the variance of the error term to depend on the explanatory variable. This flexibility leads to the second contribution: we provide a simple Bayesian approach to robustly estimate finite population means and ratios. The strategy to attain whole robustness is simple since it lies in replacing the traditional normal assumption on the error term by a super heavy-tailed distribution assumption. As a result, users can estimate the parameters as usual, using the posterior distribution.