Robust Bayesian model averaging for linear regression models with heavy-tailed errors

Area of Science:

• Statistics
• Statistical Modeling
• Bayesian Inference

Background:

• Traditional linear regression assumes normally distributed errors, which is often violated in real-world data due to outliers.
• Existing methods like the Bayesian Huberized lasso have limitations in enforcing sparsity (coefficients exactly zero).
• The hyperbolic and Student-t distributions offer alternatives to the normal distribution for modeling heavier tails, but their shapes and tail behaviors differ.

Purpose of the Study:

• To develop a Bayesian model averaging technique for linear regression that accommodates heavier-tailed error distributions.
• To propose a Bayesian variable selection approach using spike and slab priors for more effective sparsity enforcement.
• To introduce a flexible error distribution that encompasses both hyperbolic and Student-t families, with an estimated tail heaviness parameter.

Main Methods:

• Development of a Bayesian variable selection approach with spike and slab priors.
• Proposal of a flexible error distribution combining hyperbolic and Student-t characteristics.
• Implementation of an efficient Gibbs sampler for posterior computation.

Main Results:

• The proposed method demonstrates competitive performance against state-of-the-art techniques.
• Simulation studies and real dataset analyses validate the effectiveness of the new Bayesian approach.
• The model successfully handles heavier-tailed error distributions and improves variable selection accuracy.

Conclusions:

• The developed Bayesian regression model offers a flexible and effective solution for variable selection with heavier-tailed errors.
• The method provides a robust alternative to existing techniques, particularly in the presence of outliers.
• The approach enhances the ability to model complex error structures in linear regression.