High-dimensional AIC in the growth curve model

The AIC and its modifications have been proposed for selecting the degree in a polynomial growth curve model under a large-sample framework when the sample size n is large, but the dimension p is fixed. In this paper, first we propose a high-dimensional AIC (denoted by HAIC) which is an asymptotic unbiased estimator of the AIC-type risk function defined by the expected log-predictive likelihood or equivalently the Kullback–Leibler information, under a high-dimensional framework such that p/n→c∈[0,1). It is noted that our new criterion gives an estimator with small biases in a wide range of p and n. Next we derive asymptotic distributions of AIC and HAIC under the high-dimensional framework. Through a Monte Carlo simulation, we note that these new approximations are more accurate than the approximations based on a large-sample framework.