Classical and Bayesian Inference Robustness in Multivariate Regression Models

Abstract

We consider a one-to-one correspondence between points z ∈ Rⁿ - {0} and pairs (y, r), where r > 0 and y lies in some space y, through z = r y. As an immediate consequence, we can represent random variables Z that take values in Rⁿ - {0} as Z = R Y, where R is a positive random variable and Y takes values in y. By fixing the distribution of either R or Y while imposing independence between them, we generate classes of distributions on Rⁿ . Many families of multivariate distributions (e.g., spherical, elliptical, l_q spherical, v spherical, and anisotropic) can be interpreted in this unifying framework. Some classical inference procedures can be shown to be completely robust in these classes of multivariate distributions. We use these findings in the practically relevant context of regression models. Finally, we present a robust Bayesian analysis and indicate the links between classical and Bayesian results. In particular, for the regression model with iid errors up to a scale, we provide a formal characterization for both classical and Bayesian robustness results concerning inference on the regression parameters.

Key Words:

• Distribution invariance
• Independent sampling
• Likelihood ratio principle
• Pivotal quantity
• Posterior inference
• Scale invariance.