Variable Selection and Parameter Estimation with the Atan Regularization Method

Abstract

Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by penalized least squares using various penalty functions. In this paper, an arctangent type penalty which very closely resembles l 0 penalty is proposed; we call it Atan penalty. The Atan-penalized least squares procedure is shown to consistently select the correct model and is asymptotically normal, provided the number of variables grows slower than the number of observations. The Atan procedure is efficiently implemented using an iteratively reweighted Lasso algorithm. Simulation results and data example show that the Atan procedure with BIC-type criterion performs very well in a variety of settings.