Spectral analysis of large dimensional Chatterjee's rank correlation matrix

Abstract

This paper studies the spectral behavior of large dimensional Chatterjee's rank correlation matrix when observations are independent draws from a high-dimensional random vector with independent continuous components. We show that the empirical spectral distribution of its symmetrized version converges to the semicircle law, and thus providing the first example of a large correlation matrix deviating from the Marchenko-Pastur law that governs those of Pearson, Kendall, and Spearman. We further establish central limit theorems for linear spectral statistics, which in turn enable the development of Chatterjee's rank correlation-based tests of complete independence among the components.

Figures (3)

• Figure 1: Semicircle law of $\mathbf{\Phi}_n$ with $p=100$ and $n=200$.
• Figure 2: M-P law of $\bm{\Psi}_n$ with $p=100$ and $n=200$.
• Figure 3: Histograms of $\operatorname{tr}(\mathbf{\Psi}_n^k)$ centered by sample means, with $p=500$, $n=200$, and over 500 replications.

Theorems & Definitions (37)

• Theorem 1.1: Semicircle law for $\bm{\Phi}_n$
• Theorem 1.2: M-P law for $\bm{\Psi}_n$
• Theorem 1.3: CLT for LSS of $\bm{\Psi}_n$
• Proposition 2.1: MR4185806
• Remark 2.1
• Proposition 2.2: Dependence structure of relative ranks
• Corollary 2.1
• Proposition 2.3
• Proposition 2.4
• Proposition 4.1