  Non-concave penalized maximum likelihood methods, such as the Bridge, the
SCAD, and the MCP, are widely used because they not only do parameter
estimation and variable selection simultaneously but also have a high
efficiency as compared to the Lasso. They include a tuning parameter which
controls a penalty level, and several information criteria have been developed
for selecting it. While these criteria assure the model selection consistency
and so have a high value, it is a severe problem that there are no appropriate
rules to choose the one from a class of information criteria satisfying such a
preferred asymptotic property. In this paper, we derive an information
criterion based on the original definition of the AIC by considering the
minimization of the prediction error rather than the model selection
consistency. Concretely speaking, we derive a function of the score statistic
which is asymptotically equivalent to the non-concave penalized maximum
likelihood estimator, and then we provide an asymptotically unbiased estimator
of the Kullback-Leibler divergence between the true distribution and the
estimated distribution based on the function. Furthermore, through simulation
studies, we check that the performance of the proposed information criterion
gives almost the same as or better than that of the cross-validation.
