Chatterjee's Rank Correlation Matrix

• Chatterjee's rank correlation matrix is a multivariate tool that quantifies directed, possibly nonlinear functional dependence using pairwise rank measures.
• It exhibits a unique spectral behavior with a semicircular limiting law in high dimensions, contrasting with classical symmetric correlation matrices.
• The framework supports robust nonparametric global independence testing through a central limit theorem for linear spectral statistics.

Chatterjee's rank correlation matrix is a multivariate extension of Chatterjee's rank correlation coefficient, which quantifies the directed, potentially nonlinear functional dependence between random variables. Unlike classical measures of concordance—such as Pearson, Spearman, or Kendall—which are symmetric and primarily evaluate monotonic association, the Chatterjee coefficient and its matrix generalization are explicitly asymmetric and designed to detect the extent to which one variable is a measurable function of another. The matrix variant is constructed from pairwise Chatterjee rank coefficients and, when symmetrized, exhibits unique spectral behavior in high dimensions, providing a new analytical framework for dependence and independence testing in modern statistical data analysis.

1. Mathematical Definition and Construction

The Chatterjee rank correlation between two continuous random variables $X$ and %%%%1%%%% is, in copula form,

$$\xi(C) = 6\int_0^1\int_0^1 (\partial_1 C(u, v))^2 \, u v \, du dv - 2$$

where %%%%2%%%% is the copula of $(X, Y)$, and $\partial_1 C$ denotes the partial derivative with respect to $u$.

The $p \times p$ Chatterjee rank correlation matrix $\Xi_n$ for a $p$-dimensional random vector $\mathbf{X} = (X^{(1)},\ldots, X^{(p)})$ is defined entrywise as

$[\Xi_n]_{jk} = \xi_n(X^{(j)}, X^{(k)})$

where $\xi_n(\cdot, \cdot)$ is the empirical Chatterjee rank correlation calculated from the sample, i.e.,

$$\xi_n(X, Y) = 1 - \frac{3\sum_{i=1}^{n-1}|r_{i+1} - r_i|}{n^2 - 1}$$

with $r_i$ being the rank of the $i$th $Y$-observation after sorting the $X$-values.

Unlike the Pearson and Spearman correlation matrices, $\Xi_n$ is not symmetric in general, though its symmetrized variant $\Phi_n = (\Xi_n + \Xi_n^T)/2$ is often used in spectral analysis.

2. Spectral Limit Laws in High-Dimensional Regimes

The main spectral result for large-dimensional Chatterjee's rank correlation matrices concerns the limiting empirical spectral distribution (ESD) of the symmetrized matrix $\Phi_n$ when the data are high-dimensional with independent continuous components: $$\frac{1}{p} \sum_{i=1}^p \delta_{\lambda_i(\Phi_n)} \xrightarrow{a.s.} \mu_{sc}$$ as $n, p \to \infty$ with $p/n \to \gamma \in (0, \infty)$. The limiting measure $\mu_{sc}$ is the Wigner semicircle law with density

$$\rho_{sc}(x) = \frac{1}{2\pi \sigma^2} \sqrt{4\sigma^2-x^2} \ \mathbb{I}(|x|\leq 2\sigma),$$

where for the Chatterjee matrix, $\sigma^2 = 2\gamma/5$ (Dong et al., 8 Oct 2025).

This is fundamentally different from the Marchenko–Pastur law, which governs empirical spectral distributions of classical correlation matrices (Pearson, Spearman, Kendall, etc.), indicating distinctive spectral universality for rank-based dependence measures.

3. Central Limit Theorems for Linear Spectral Statistics

For analytic functions $f$, linear spectral statistics of the form

$$L_n(f) = \sum_{i=1}^p f(\lambda_i) - p\int f(x)\,d\mu_{sc}(x)$$

obey a central limit theorem: $L_n(f) \xrightarrow{d} \mathcal{N}(0,V(f)),$ where $V(f)$ is an explicit variance depending on moments of $f$ and $\gamma$ (Dong et al., 8 Oct 2025). For polynomial test functions $f(x) = x^k$, the covariance structure of the limiting Gaussian process involves combinatorial quantities (e.g., Catalan numbers) and detailed "graph-counting" reflecting local rank dependence.

This statistical machinery legitimizes hypothesis testing for global independence based on the aggregate spectral features of the Chatterjee rank matrix, as opposed to elementwise thresholding or maximum statistics.

4. Practical Implications: Nonparametric Testing for Independence

The spectral results imply a new class of high-dimensional nonparametric independence tests. A generic test statistic can be constructed as

$$Q(f) = \frac{\operatorname{tr}(\Psi_n^k) - \mathbb{E}\operatorname{tr}(\Psi_n^k)}{\sqrt{\operatorname{Var}\left[\operatorname{tr}(\Psi_n^k)\right]}}$$

where $\Psi_n$ is a suitably centered version of the (symmetrized) Chatterjee rank matrix, and the denominator is calculated using the variance established by the CLT for linear spectral statistics.

Under the null of complete independence, $Q(f) \to_d N(0,1)$. This provides a nonparametric alternative to likelihood-based (Gaussian-centric) or classical rank-based global tests and is robust to nonlinearity, non-monotonicity, and heavy-tailed marginals.

5. Distinctive Features Versus Classical Correlation Matrices

Matrix Type	Limiting Spectrum	Asymmetry	Sensitivity	Spectral Parameter
Pearson	Marchenko–Pastur	Symmetric	Linear dependence	Population variance
Spearman/Kendall	Marchenko–Pastur	Symmetric	Monotonic relationships	Population variance
Chatterjee	Wigner Semicircle	Asymmetric	Functional dependence	$2\gamma/5$

Chatterjee’s matrix is uniquely asymmetric (since $\xi(X,Y) \ne \xi(Y,X)$ in general), functionally oriented (detecting whether $Y$ is determined by $X$), and its spectral limit is governed by a semicircle law—highlighting fundamentally different aggregate dependence behavior in large systems (Dong et al., 8 Oct 2025).

6. Theoretical and Methodological Developments

The establishment of the semicircle law for the empirical spectral distribution arises from the combinatorial structure of the rank coefficient, the asymptotic normality of its entries (variance $2/5$ under the null), and specific moment method calculations for the symmetrized matrix.

The proof relies on detailed analysis of the weak convergence of linear spectral statistics, utilizing techniques from random matrix theory, combinatorial enumeration, and the asymptotic properties of rank-based $U$- and $V$-statistics.

7. Future Directions and Open Problems

• Random matrix universality: Whether versions of the semicircle law persist for other rank-based or asymmetric dependence measures with different marginal structures.
• Depiction under dependence: The behavior of the ESD and linear spectral statistics when the underlying variables are dependent or in the presence of ties.
• Applications beyond independence testing: Use in graphical modeling, covariance estimation under model misspecification, and as a diagnostic for functional nonlinearity.
• Extension to multivariate and robust variants: Construction and theory for extensions incorporating multivariate rank correlations (such as Azadkia–Chatterjee–type graphs) and rank-based causal discovery.

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Chatterjee's rank correlation matrix thus stands as a new, theoretically grounded tool for quantifying and testing high-dimensional dependence and functional association, with a spectral distribution and fluctuation theory fundamentally departing from those of classical symmetric correlation matrices (Dong et al., 8 Oct 2025). Its main properties—semicircle spectral limit, robust CLT for linear statistics, and directed functional sensitivity—enable principled, distribution-free methodologies for modern multivariate statistical analysis.