Consistency Properties of Model Selection Criteria in Multiple Linear Regression

This paper concerns the asymptotic properties of a class of criteria for model selection in linear regression models, which covers the most well known criteria as e.g. MALLOWS' Cp, CV (cross-validation), GCV ( generalized cross-validation), AKAIKE's AIC and FPE as well as SCHWARZ' BIC. These criteria are shown to be consistent in the sense of selecting the true or larger models, assuming i.i.d. errors and the possible inadequacy of the linear model. Additionally we prove that BIC-type criteria select the true model if the sample size is converging to infinity. These consistency properties are completed by convergence results for the risk and loss of the estimated regression functions.