  This paper proposes an adaptive penalized weighted mean regression for
outlier detection of high-dimensional data. In comparison to existing
approaches based on the mean shift model, the proposed estimators demonstrate
robustness against outliers present in both response variables and/or
covariates. By utilizing the adaptive Huber loss function, the proposed method
is effective in high-dimensional linear models characterized by heavy-tailed
and heteroscedastic error distributions. The proposed framework enables
simultaneous and collaborative estimation of regression parameters and outlier
detection. Under regularity conditions, outlier detection consistency and
oracle inequalities of robust estimates in high-dimensional settings are
established. Additionally, theoretical robustness properties, such as the
breakdown point and a smoothed limiting influence function, are ascertained.
Extensive simulation studies and a breast cancer survival data are used to
evaluate the numerical performance of the proposed method, demonstrating
comparable or superior variable selection and outlier detection capabilities.
