Condition number-based new robust ridge M-estimator for linear regression model with penalty of multicollinearity and outlier

In multiple linear regression models, traditional methods, such as OLS and ridge regression often fail due to multicollinearity and outliers in the y-direction. To address these issues, robust ridge M-estimators have been developed, offering resistance to outliers and multicollinearity. Selecting an optimal ridge parameter helps minimize mean squared error (MSE). However, many existing methods are ineffective when both multicollinearity and outliers coexist. This study proposes a new robust ridge M-estimator that utilizes the MSE criteria to outperform previous approaches. The efficiency of the proposed estimator is evaluated through simulation studies. It is recommended for use in scenarios with varying degrees of multicollinearity and noise, as it automatically adapts to these conditions. This estimator performs well in simulations involving high error variance, y-direction outliers, and strong multicollinearity. The efficiency of the estimator is demonstrated under both normal and heavy-tailed error distributions, and an empirical example further confirms its practical effectiveness.