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Abstract
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.