M-estimation in high-dimensional linear model

We mainly study the M-estimation method for the high-dimensional linear regression model and discuss the properties of the M-estimator when the penalty term is a local linear approximation. In fact, the M-estimation method is a framework which covers the methods of the least absolute deviation, the quantile regression, the least squares regression and the Huber regression. We show that the proposed estimator possesses the good properties by applying certain assumptions. In the part of the numerical simulation, we select the appropriate algorithm to show the good robustness of this method.