  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 of symmetry and homogeneity, or the
covariates are not sub-Gaussian. To reduce the possible bias caused by the
data's irregularity in mean regression, PRAM adopts a loss function with a
flexible robustness parameter growing with the sample size. 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 demonstrate the performances of six PRAM estimators using
three types of loss functions for approximation (Huber, Tukey's biweight and
Cauchy loss) combined with two types of penalty functions (Lasso and MCP). Our
simulation studies and real data analysis demonstrate satisfactory finite
sample performances of the PRAM estimator under general irregular settings.
