  It has been shown that AIC-type criteria are asymptotically efficient
selectors of the tuning parameter in non-concave penalized regression methods
under the assumption that the population variance is known or that a consistent
estimator is available. We relax this assumption to prove that AIC itself is
asymptotically efficient and we study its performance in finite samples. In
classical regression, it is known that AIC tends to select overly complex
models when the dimension of the maximum candidate model is large relative to
the sample size. Simulation studies suggest that AIC suffers from the same
shortcomings when used in penalized regression. We therefore propose the use of
the classical corrected AIC (AICc) as an alternative and prove that it
maintains the desired asymptotic properties. To broaden our results, we further
prove the efficiency of AIC for penalized likelihood methods in the context of
generalized linear models with no dispersion parameter. Similar results exist
in the literature but only for a restricted set of candidate models. By
employing results from the classical literature on maximum-likelihood
estimation in misspecified models, we are able to establish this result for a
general set of candidate models. We use simulations to assess the performance
of AIC and AICc, as well as that of other selectors, in finite samples for both
SCAD-penalized and Lasso regressions and a real data example is considered.
