Consistency properties of AIC, BIC, Cp and their modifications in the growth curve model under a large-(q,n) framework

Abstract

This paper is concerned with consistency properties of some criteria for selecting row vectors of a $k\times p$ design matrix within individuals in the growth curve model, based on a sample of size $n$. Recently Enomoto, Sakurai and Fujikoshi (2013) showed that $\text{AIC}$ and its modification have a consistency property for selecting hierarchical models of the row vectors under a condition on the order of the noncentrality matrix, assuming a large-$(q,n)$ asymptotic framework such that $q/n\to d\in [0,1)$. We extend the result to a family of log-likelihood-based information criteria including $\text{AIC}$ and $\text{BIC}$, and $\text{Cp}$. Further, their consistency properties are also obtained under a new condition on the order of the noncentrality matrix. Our results are checked numerically by conducting a Mote Carlo simulation.