Maximum likelihood principle and model selection when the true model is unspecified (Q1825556)

Classical results on this topic are based on the assumption that the unknown density function lies in a specific parametric family. What happens when this assumption does not hold? In addressing this question the author considers the maximum likelihood based on a specified parametric family which provides a good approximation, in a specified sense, to the true distribution. Asymptotic properties of the MLE and of the maximum likelihood are explored, and the results are applied to the problem of model selection, the BIC and AIC criteria being shown to be strongly consistent and inconsistent, respectively.