Adaptive LASSO based on joint M-estimation of regression and scale

The adaptive Lasso (Least Absolute Shrinkage and Selection Operator) obtains oracle variable selection property by using cleverly chosen adaptive weights for regression coefficients in the ℓ1-penalty. In this paper, in the spirit of M-estimation of regression, we propose a class of adaptive M-Lasso estimates of regression and scale as solutions to generalized zero subgradient equations. The defining estimating equations depend on a differentiable convex loss function and choosing the LS-loss function yields the standard adaptive Lasso estimate and the associated scale statistic. An efficient algorithm, a generalization of the cyclic coordinate descent algorithm, is developed for computing the proposed M-Lasso estimates. We also propose adaptive M-Lasso estimate of regression with preliminary scale estimate that uses a highly-robust bounded loss function. A unique feature of the paper is that we consider complex-valued measurements and regression parameter. Consistent variable selection property of the adaptive M-Lasso estimates are illustrated with a simulation study.