Limiting spectral distribution of large dimensional Spearman's rank correlation matrices

TL;DR

This paper investigates the spectral properties of Spearman's rank correlation matrices in high dimensions, revealing that their limiting distribution follows a generalized Mare7enko-Pastur law, and compares them with other classical matrices.

Contribution

It establishes the limiting spectral distribution of Spearman's rank correlation matrices as a generalized Mare7enko-Pastur law, extending spectral analysis to rank-based correlations.

Findings

Limiting spectral distribution follows a generalized Mare7enko-Pastur law.

Spearman's correlation matrices behave similarly to other classical matrices in high dimensions.

Comparison of Spearman's, Pearson's, Kendall's, and sample covariance matrices.

Abstract

In this paper, we study the empirical spectral distribution of Spearman's rank correlation matrices, under the assumption that the observations are independent and identically distributed random vectors and the features are correlated. We show that the limiting spectral distribution is the generalized Mar\u{c}enko-Pastur law with the covariance matrix of the observation after standardized transformation. With these results, we compare several classical covariance/correlation matrices including the sample covariance matrix, Pearson's correlation matrix, Kendall's correlation matrix and Spearman's correlation matrix.