Bayesian information criteria for multiple regression models: A study of robustness and comparisons

It is recommended to employ robust versions of selection criteria for variables in linear regression when extreme outliers and high leverage points present. This article investigates the robust version of the Bayesian information criterion based on the <i>MM</i>-estimation scale <i>BIC_MM</i>. Furthermore, the robustness properties of the proposed <i>BIC_MM</i> criterion are examined <i>via</i> its breakdown point, influence function, and gross-error sensitivity. Performance of the proposed criterion is compared to existing non-robust and robust Bayesian information criterion, robust Akaike’s information criterion, robust Mallows’ Cp, and robust Schwarz information criterion, based on the least-trimmed squares estimator. Simulation studies and applications on real data reveal that <i>BIC_MM</i> selects more suitable models in situations where there are outliers or leverage points. Consequently, when multicollinearity exists, the proposed robust <i>BIC_MM</i> can be used to select the tuning parameter in shrinkage regression methods such as <i>ridge</i> and <i>lasso</i> regression.