  Fan and Li propose a family of variable selection methods via penalized
likelihood using concave penalty functions. The nonconcave penalized likelihood
estimators enjoy the oracle properties, but maximizing the penalized likelihood
function is computationally challenging, because the objective function is
nondifferentiable and nonconcave. In this article, we propose a new unified
algorithm based on the local linear approximation (LLA) for maximizing the
penalized likelihood for a broad class of concave penalty functions.
Convergence and other theoretical properties of the LLA algorithm are
established. A distinguished feature of the LLA algorithm is that at each LLA
step, the LLA estimator can naturally adopt a sparse representation. Thus, we
suggest using the one-step LLA estimator from the LLA algorithm as the final
estimates. Statistically, we show that if the regularization parameter is
appropriately chosen, the one-step LLA estimates enjoy the oracle properties
with good initial estimators. Computationally, the one-step LLA estimation
methods dramatically reduce the computational cost in maximizing the nonconcave
penalized likelihood. We conduct some Monte Carlo simulation to assess the
finite sample performance of the one-step sparse estimation methods. The
results are very encouraging.
