Consistency of high-dimensional AIC-type and CpCp-type criteria in multivariate linear regression

The AIC, the multivariate Cp and their modifications have been proposed for multivariate linear regression models under a large-sample framework when the sample size nn is large, but the dimension pp of the response variables is fixed. In this paper, first we propose a high-dimensional AIC (denoted by HAIC) which is an asymptotic unbiased estimator of the risk function defined by the expected log-predictive likelihood or equivalently the Kullback–Leibler information under a high-dimensional framework p/n→c∈[0,1)p/n→c∈[0,1). It is noted that our new criterion provides better approximations to the risk function in a wide range of pp and nn. Recently Yanagihara et al. (2012)  [17] noted that AIC has a consistency property under Ω=O(np) when p/n→c∈[0,1)p/n→c∈[0,1), where Ω is a noncentrality matrix. In this paper we show that several criteria including HAIC and Cp have also a consistency property under Ω=O(n) as well as Ω=O(np) when p/n→c∈[0,1)p/n→c∈[0,1). Our results are checked numerically by conducting a Monte Carlo simulation.