Global optimality of nonconvex penalized estimators

Nonconvex penalties such as the smoothly clipped absolute deviation or minimax concave penalties have desirable properties such as the oracle property, even when the dimension of the predictive variables is large. However, checking whether a given local minimizer has such properties is not easy since there can be many local minimizers. In this paper, we give sufficient conditions under which a local minimizer is unique, and show that the oracle estimator becomes the unique local minimizer with probability tending to one. Copyright 2012, Oxford University Press.