On the underfitting and overfitting sets of models chosen by order selection criteria

For a general class of order selection criteria, we establish analytic and non-asymptotic evaluations of both the underfitting and overfitting sets of selected models. These evaluations are further specified in various situat ions including regressions and autoregressions with finite or infinite vari ances. We also show how upper bounds for the misfitting probabilities and h ence conditions ensuring the weak consistency can be derived from the given evaluations. Moreover, it is demonstrated how these evaluations, combined with a law of the iterated logarithm for some relevant statistic, can provi de conditions ensuring the strong consistency of the model selection criter ion used. (C) 1999 Academic Press.