Consistency of AIC and its modification in the growth curve model under a large-(q,n) framework

The AIC and its modifications have been proposed for selecting the degree in a polynomial growth curve model under a large-sample framework and a high-dimensional framework by Satoh, Kobayashi and Fujikoshi [9] and Fujikoshi, Enomoto and Sakurai [4], respectively. They note that the AIC and its modifications have no consistency property. In this paper we consider asymptotic properties of the AIC and its modification when the number q of groups or explanatory variables and the sample size n are large. First we show that the AIC has a consistency property under a large-(q, n) framework such that q/n → d ∈ [0, 1), under a condition on the noncentrality matrix, but the dimension p is fixed. Next we propose a modification of the AIC (denoted by MAIC) which is an asymptotic unbiased estimator of the risk under the asymptotic framework. It is shown that MAIC has a consistency property under a condition on the noncentrality matrix. Our results are checked numerically by conducting a Mote Carlo simulation. AMS 2010 Mathematics Subject Classification. 62H12, 62H30.