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

The AIC , the multivariate C p and their modifications have been proposed for multivariate linear regression models under a large-sample framework when the sample size n is large, but the dimension p 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 ) . It is noted that our new criterion provides better approximations to the risk function in a wide range of p and n . Recently Yanagihara et al. (2012)  [17] noted that AIC has a consistency property under Ω = O ( n p ) when p / n → c ∈ [ 0 , 1 ) , where Ω is a noncentrality matrix. In this paper we show that several criteria including HAIC and C p have also a consistency property under Ω = O ( n ) as well as Ω = O ( n p ) when p / n → c ∈ [ 0 , 1 ) . Our results are checked numerically by conducting a Monte Carlo simulation.