Abstract
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.
Highlights
For the classical linear regression model Y = Xβ + ε, we are interested in the problem of variable selection and estimation, where Y = (y1, y2, ..., yn)T is the response vector, X = (X1, X2, ..., Xpn) = (x1, x2, ..., xn)T =n×pn is an n × pn design matrix, and ε = (ε1, ε2, ..., εn)T is a random vector
The main topic is how to estimate the coefficients vector β ∈ Rpn when pn increases with sample size n and many elements of β equal zero. We can transfer this problem into a minimization of a penalized least squares objective function pn βn arg min β
LAD estimation is the special case of M-estimation, which is named by Huber(1964, 1973, 1981) [1] [2] [3]firstly and can be obtained by minimizing the objective function
Summary
The main topic is how to estimate the coefficients vector β ∈ Rpn when pn increases with sample size n and many elements of β equal zero. We can transfer this problem into a minimization of a penalized least squares objective function pn βn. |yi − xTi βn| + pλn(|βnj|), i=1 j=1 where the loss function is least absolute deviation(LAD for short), that does not need the noise obeys a gaussian distribution and be more robust than least squares estimation. LAD estimation is the special case of M-estimation, which is named by Huber(1964, 1973, 1981) [1] [2] [3]firstly and can be obtained by minimizing the objective function.
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