Title: High-Dimensional Properties of IC for Selection of Reduced Ranks in Sparse Multivariate Linear Model
Abstract: First, we consider IC (Information Criteria) for selection of reduced ranks in multivariate regression model with \(p\) response variables, \(q\) explanatory variables and \(n\) samples when the covariance matrix of response variables is \(\Sigma = \sigma^2 I_p\). Sufficient conditions are given for IC to be consistent in a high-dimensional situation when \(p/n \to c >0\). We also consider IC for selecting both reduced ranks and explanatory variables. Two different approaches, due to Bunea, She and Wegkamp (AS; 2011, 2012) and Chen and Huang (JASA; 2012), are reviewed.
Next, the results are extended to IC for estimating the number of significant discriminant functions in multiple discriminant analysis. We also consider the case where \(\Sigma\) is positive definite, based on Fujikoshi and Sakurai (JMA; 2016).