Asymptotic Overfitting Probabilities of some Model Selection Criteria in Large Samples

We look at overfitting in large samples by obtaining asymptotic probabilities of some model selection criteria, which are AIC, AICc, AICu, SIC, HQ, HQc, Cp, FPE, and FPEu. Asymptotic Overfitting probabilities refer to comparing one overfitted model of order k_{*} +L to the reduced model of order k_{*}. We make use of the following fact : As n rarrow INF and where k_{*} and L are fixed, {chi_{ n-k_{*} -L}^{2}} over { n-k_{*} -L} rarrow 1. It is clear that AIC, AICc, FPE and Cp are not constant criteria, since their probabilities of overfitting are not 0. But SIC, HQ, and HQc are consistent and thus do have 0 probability of overfitting.