High-Dimensional Consistency of Estimation Criteria for the Rank in Multivariate Linear Model

The model selection criteria AIC, BIC and Cp have been proposed for estimation of the rank of coefficient matrix in multivariate linear model. In general, it is known that under a large-sample asymptotic framework AIC and Cp is not consistent, but BIC is consistent. However, we note that these criteria have consistency when the number p of the response variables and the sample size n are large under a highdimensional asymptotic framework such that the ratio p/n tends to a constant c (0 ≤ c < 1) as p and n are large. The consistency properties are also shown for extended criteria with a tuning parameter. Further, we propose the ridge-type criteria whose justifications are given under a large-sample asymptotic framework. Their consistencies are shown in a high-dimensional asymptotic framework. Through a Monte Carlo simulation experiment our results are checked numerically, and the estimation criteria are compared. AMS 2000 subject classification: primary 62H12; secondary 62H30