High-dimensional consistency of rank estimation criteria in multivariate linear model

This paper is concerned with consistency properties of rank estimation criteria in a multivariate linear model, based on the model selection criteria AIC, BIC and Cp. The consistency properties of these criteria are studied under a high-dimensional framework with two different assumptions on the noncentrality matrix such that the number of response variables and the sample size tend to infinity. In general, it is known that under a large-sample asymptotic framework, the criteria based on AIC and Cp are not consistent, but the criterion based on BIC is consistent. However, we note that there are cases that the criteria based on AIC and Cp are consistent, but the criterion based on BIC is not consistent. Such consistency properties are also shown for the generalized criteria with a tuning parameter. Further, the modified criteria with a ridge-type estimator are also examined. Through a Monte Carlo simulation experiment, our results are checked numerically, and the estimation criteria are compared.