Overview
This study investigates the spectral behavior of large dimensional Chatterjee's rank correlation matrix when observations are drawn independently from a high-dimensional random vector characterized by independent continuous components. The research establishes limiting empirical spectral distributions (ESDs) for two symmetrized forms of this matrix within a proportional high-dimensional regime. One of these limiting distributions is identified as the semicircle law. Additionally, the paper develops central limit theorems (CLTs) for linear spectral statistics (LSS) associated with these symmetrized matrices.
Research Context
The spectral properties of correlation matrices are a recognized area of study in high-dimensional statistics. Prior research has identified specific spectral limits, such as the Marchenko–Pastur law, for matrices derived from Pearson, Kendall, and Spearman correlations. This work contributes to this context by examining Chatterjee's rank correlation matrix, offering a comparative spectral analysis. Specifically, the observed semicircle law for one symmetrized version represents a divergence from the Marchenko–Pastur spectral limit seen in other established correlation types.
Approach
The methodology centers on analyzing the spectral characteristics of Chatterjee's rank correlation matrix under specific conditions: large dimensionality and independent draws from a high-dimensional random vector with independent continuous components. The analysis focuses on two distinct symmetrized versions of the matrix. The key analytical steps include:
- Establishing limits for the empirical spectral distributions of these two symmetrized versions.
- Identifying specific functional forms for these limiting distributions, such as the semicircle law.
- Deriving central limit theorems for linear spectral statistics of the symmetrized matrices.
The study operates within a proportional high-dimensional regime, a common framework for analyzing large dimensional random matrices.
Findings
The research yielded several specific findings regarding the spectral behavior of large dimensional Chatterjee's rank correlation matrix:
- Limits for the empirical spectral distributions were established for two symmetrized versions of the matrix.
- One of the symmetrized versions was found to exhibit the semicircle law as its spectral limit.
- This finding distinguishes Chatterjee's correlation matrix from Pearson, Kendall, and Spearman cases, which typically exhibit a Marchenko–Pastur spectral limit.
- Central limit theorems were established for linear spectral statistics of both symmetrized matrices.
Potential Applications
An important application of the developed theory is in creating Chatterjee's rank correlation-based tests. These tests can be used to assess the complete independence among the components of a high-dimensional random vector.