A consistency property of the AIC for multivariate linear models when the dimension and the sample size are large

Abstract

It is common knowledge that Akaike’s information criterion (AIC) is not a consistent model selection criterion, and Bayesian information criterion (BIC) is. These have been confirmed from an asymptotic selection probability evaluated from a large-sample framework. However, when a high-dimensional asymptotic framework, such that the dimension of the response variables and the sample size are approaching $\infty$, is used for evaluating the selection probability, there are cases that the AIC for selecting variables in multivariate linear models is consistent, but the BIC is not. The AIC and BIC are included in a family of information criteria defined by adding a penalty term expressing the complexity of the model to a negative twofold maximum log-likelihood. By clarifying the condition of the penalty term to ensure the consistency, we derive conditions for consistency of the AIC, BIC and other information criteria under the high-dimensional asymptotic framework.