High-dimensional robust approximated M-estimators for mean regression with asymmetric data

Asymmetry along with heteroscedasticity or contamination often occurs with the growth of data dimensionality. In ultra-high dimensional data analysis, such irregular settings are usually overlooked for both theoretical and computational convenience. In this paper, we establish a framework for estimation in high-dimensional regression models using Penalized Robust Approximated quadratic M-estimators (PRAM). This framework allows general settings such as random errors lack symmetry and homogeneity, or covariates are not sub-Gaussian. To reduce the possible bias caused by data’s irregularity in mean regression, PRAM adopts a loss function with an adaptive robustification parameter. Theoretically, we first show that, in the ultra-high dimension setting, PRAM estimators have local estimation consistency at the minimax rate enjoyed by the LS-Lasso. Then we show that PRAM with an appropriate non-convex penalty in fact agrees with the local oracle solution, and thus obtain its oracle property. Computationally, we compare the performances of six PRAM estimators (Huber, Tukey’s biweight or Cauchy loss function combined with Lasso or MCP penalty function). Our simulation studies and real data analysis demonstrate satisfactory finite sample performances of the PRAM estimator under different irregular settings.