A robust and sparse estimation technique for linear regression problem is studied in this paper. Standard regression with Lasso, SCAD and MCP regularizations is not robust against outliers since it involves the least squares. To handle outliers, a two-stage procedure is proposed; at the first stage an initial estimator is calculated and then it is improved at the second stage by iteratively solving a sparse regression problem with reducing outlier effects. This procedure includes not only a random error but also a computational error. The convergence performance for the final estimator is investigated in both computational and statistical perspectives.
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