The L[subscript 1] penalized LAD estimator for high dimensional linear regression

Abstract

In this paper, the high-dimensional sparse linear regression model is considered, where the overall number of variables is larger than the number of observations. We investigate the L[subscript 1] penalized least absolute deviation method. Different from most of the other methods, the L[subscript 1] penalized LAD method does not need any knowledge of standard deviation of the noises or any moment assumptions of the noises. Our analysis shows that the method achieves near oracle performance, i.e. with large probability, the L[subscript 2] norm of the estimation error is of order View the O(√k log p/n). The result is true for a wide range of noise distributions, even for the Cauchy distribution. Numerical results are also presented.