High-dimensional robust approximated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e725" altimg="si2.svg"><mml:mi>M</mml:mi></mml:math>-estimators for mean regression with asymmetric data

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