High-dimensional properties of AIC, BIC and Cp for estimation of dimensionality in canonical correlation analysis

Abstract

This paper is concerned with consistency properties of the dimensionality estimation criteria AIC, BIC, and ${\text{C}}_p$ in CCA (Canonical Correlation Analysis) between $p$ variables and $q(\le p)$ variables, based on a sample of size $N=n+1$. The consistency properties of the criteria are studied under a high-dimensional asymptotic framework such that $p$ and $n$ tend to infinity satisfying $p/n\to c\in [0,1)$, and under two types of assumptions on the order of the population canonical correlations, where $q$ is fixed. We note that there are cases that the criteria based on AIC and ${\text{C}}_p$ are consistent, but the criterion based on BIC is not consistent. Through a Monte Carlo simulation experiment, our results are checked numerically, and the estimation criteria are compared.