This is the first book on an evaluation of (weak) consistency of an information criterion for variable selection in high-dimensional multivariate linear regression models by using the high-dimensional asymptotic framework. It is an asymptotic framework such that the sample size n and the dimension of response variables vector p are approaching 8 simultaneously under a condition that p/n goes to a constant included in [ 0,1).Most statistical textbooks evaluate consistency of an information criterion by using the large-sample asymptotic framework such thatn goes to 8 under the fixed p. The evaluation of consistency of an information criterion from the high-dimensional asymptotic framework provides new knowledge to us, e.g., Akaike's information criterion (AIC) sometimes becomes consistent under the high-dimensional asymptotic framework although it never has a consistency under the large-sample asymptotic framework; and Bayesian information criterion (BIC) sometimes becomes inconsistent under the high-dimensional asymptotic framework although it is always consistent under the large-sample asymptotic framework. The knowledge may help to choose an information criterion to be used for high-dimensional data analysis, which has been attracting the attention of many researchers.
1. Introduction.-
2. Information criteria in multivariate linear regression models.- 3.Several lemmas for proving consistency.-
4. Conditions to ensure consistency for AIC-type criterion under normality.-
5. Conditions to ensure consistency for AIC-type criterion under nonnormality.-
6. Conditions to ensure consistency of Cp-type criterion under normality.-
7. Conditions to ensure consistency of Cp-type criterion under nonnormality.-
8. Appendix.
