The limiting spectral distribution of the sample canonical correlation matrix

TL;DR

This paper analyzes the spectral properties of the sample canonical correlation matrix under the alternative hypothesis, establishing its limiting spectral distribution and universality, supported by simulations.

Contribution

It introduces the limiting spectral distribution of the SCC matrix under general conditions and proves the universality of its Stieltjes transform, extending previous results.

Findings

The LSD of the SCC matrix is characterized under arbitrary distributions.

The universality of the Stieltjes transform is established.

Simulations confirm the theoretical predictions.

Abstract

In this paper, we investigate the spectral properties of the sample canonical correlation (SCC) matrix under the alternative hypothesis to provide a more comprehensive description of the association between two sets of variables. Our research involves establishing the relationship between the eigenvalues of the SCC matrix and the block correlation matrix, as well as proving the universality of the Stieltjes transform of the limiting spectral distribution (LSD) of the block correlation matrix. By combining the results from the normal case, we establish the limiting spectral distribution (LSD) of the SCC matrix with a general underlying distribution under the arbitrary rank alternative hypothesis. Finally, we present several simulated examples and find that they fit well with our theoretical results.