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An asymptotic property of model selection criteria

Probability models are estimated by use of penalized likelihood criteria related to the Akaike (1972) information criteria (AIC) and the minimum description length (MDL). The asymptotic risk of the density estimator is determined, under conditions on the penalty term, and is shown to be minimax optimal. As an application, we show that the optimal rate of convergence is achieved for the density in certain smooth nonparametric families without knowing the smooth parameters in advance.