Asymptotic non-null distributions of test statistics for redundancy in high-dimensional canonical correlation analysis

In this paper, we derive asymptotic non-null distributions of three test statistics, i.e., the likelihood ratio criterion, the Lawley-Hotelling criterion and the Bartlett-Nanda-Pillai criterion, for redundancy in high-dimensional canonical correlation analysis. Since our setting is that the dimension of one of two observation vectors may be large but does not exceed the sample size, we use the high-dimensional asymptotic framework such that the sample size and the dimension divided by the sample size tend to ∞ and some positive constant being included in [0, 1), respectively, for evaluating asymptotic distributions. Additionally, we derive asymptotic null distributions of the three test statistics under the high-dimensional asymptotic framework by using the same transformation as when we derive asymptotic non-null distribution. We verified that new approximations based on derived asymptotic distributions are more accurate than those based on asymptotic distributions evaluated from the classical asymptotic framework, i.e., only the sample size tends to ∞.