An Adaptive Generalized M-Ridge Estimator with Bounded Influence for Contaminated Multicollinear Data

Linear regression estimation can be adversely affected by multicollinearity and outliers in both the response and predictor variables. Ridge regression addresses multicollinearity but remains sensitive to outliers, whereas robust M-estimation reduces the influence of response outliers without directly addressing multicollinearity. Existing robust ridge estimators generally rely on fixed loss functions and plug-in shrinkage parameters, limiting their adaptability to diverse data structures. This paper proposes ATLAS-Ridge, a data-adaptive penalized generalized M-estimator that estimates the loss-function parameters from the data, assigns observation-specific robustness weights using robust distances, and selects the ridge parameter through generalized cross-validation. The proposed method aims to combine robust initialization, adaptive weighting, and ridge regularization within a unified estimation framework. A Monte Carlo study comprising 1408 design configurations, with 1000 replications per configuration, shows that ATLAS-Ridge attains the smallest mean squared error in 79.5% of contaminated configurations. Applications to four real datasets further demonstrate its coefficient stability. With tuning parameters fixed at their full-sample values, removing bad-leverage observations produces coefficient changes 2.7 to 5.8 times smaller than those of the best-performing competitor on three datasets and smaller than those of all competing estimators on the fourth. However, predictive performance is not uniformly superior: cross-validated prediction error ranges from a tie for first place to last among the eleven methods considered. Overall, these results suggest that ATLAS-Ridge offers a promising approach to robust linear regression under multicollinearity, although its relative performance depends on the contamination setting and evaluation criterion.