Consistency of high-dimensional AIC-type and Cp-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 @W=O(np) when p/n->c@?[0,1), where @W is a noncentrality matrix. In this paper we show that several criteria including HAIC and C"p have also a consistency property under @W=O(n) as well as @W=O(np) when p/n->c@?[0,1). Our results are checked numerically by conducting a Monte Carlo simulation.

[1]  Yasunori Fujikoshi,et al.  High-dimensional asymptotic expansions for the distributions of canonical correlations , 2009, J. Multivar. Anal..

[2]  C. L. Mallows.  Some comments on C_p , 1973 .

[3]  C. Mallows.  Some Comments on Cp , 2000, Technometrics.

[4]  Hirokazu Yanagihara,et al.  An unbiased Cp criterion for multivariate ridge regression , 2010, J. Multivar. Anal..

[5]  Y. Fujikoshi,et al.  Asymptotic expansions of the distributions of MANOVA test statistics when the dimension is large , 2014 .

[6]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[7]  Chih-Ling Tsai,et al.  MODEL SELECTION FOR MULTIVARIATE REGRESSION IN SMALL SAMPLES , 1994 .

[8]  Yasunori Fujikoshi,et al.  A criterion for variable selection in multiple discriminant analysis , 1983 .

[9]  F. J. Wyman,et al.  A comparison of asymptotic error rate expansions for the sample linear discriminant function , 1990, Pattern Recognit..

[10]  Michel J. G. Weber.  On small deviations of stationary Gaussian processes and related analytic inequalities , 2011, Sankhya A.

[11]  R. Shibata.  Selection of the order of an autoregressive model by Akaike's information criterion , 1976 .

[12]  Y. Fujikoshi,et al.  High-dimensional AICs for selection of variables in discriminant analysis , 2013, Sankhya A.

[13]  R. Sparks,et al.  The multivariate Cp , 1983 .

[14]  Y. Fujikoshi,et al.  Modified AIC and Cp in multivariate linear regression , 1997 .

[15]  James R. Schott,et al.  Testing for complete independence in high dimensions , 2005 .

[16]  H. Yanagihara.  Conditions for Consistency of a Log-Likelihood-Based Information Criterion in Normal Multivariate Linear Regression Models under the Violation of the Normality Assumption , 2015 .