An asymptotic property of model selection criteria

Probability models are estimated by use of penalized log-likelihood criteria related to AIC and MDL, The accuracies of the density estimators are shown to be related to the tradeoff between three terms: the accuracy of approximation, the model dimension, and the descriptive complexity of the model classes, The asymptotic risk is determined under conditions on the penalty term, and is shown to be minimax optimal for some cases. As an application, we show that the optimal rate of convergence is simultaneously achieved for log-densities in Sobolev spaces W-2(s)(U) without knowing the smoothness parameter s and norm parameter U in advance, Applications to neural network models and sparse density function estimation are also provided.