# Detection of Block-Exchangeable Structure in Large-Scale Correlation Matrices

## TL;DR

This paper introduces a robust method using Kendall's rank correlation to detect block structures in large correlation matrices, especially when variables form exchangeable clusters, improving estimation accuracy in high-dimensional data.

## Contribution

It proposes a new algorithm for identifying exchangeable clusters in correlation matrices without prior knowledge of the number of clusters, enhancing estimation in high-dimensional settings.

## Key findings

- The estimator outperforms sample Kendall correlation when K < d.
- It is more efficient in finite samples even when K = d.
- The method extends to linear correlations under elliptical distributions.

## Abstract

Correlation matrices are omnipresent in multivariate data analysis. When the number d of variables is large, the sample estimates of correlation matrices are typically noisy and conceal underlying dependence patterns. We consider the case when the variables can be grouped into K clusters with exchangeable dependence; this assumption is often made in applications, e.g., in finance and econometrics. Under this partial exchangeability condition, the corresponding correlation matrix has a block structure and the number of unknown parameters is reduced from d(d-1)/2 to at most K(K+1)/2. We propose a robust algorithm based on Kendall's rank correlation to identify the clusters without assuming the knowledge of K a priori or anything about the margins except continuity. The corresponding block-structured estimator performs considerably better than the sample Kendall rank correlation matrix when K < d. The new estimator can also be much more efficient in finite samples even in the unstructured case K = d, although there is no gain asymptotically. When the distribution of the data is elliptical, the results extend to linear correlation matrices and their inverses. The procedure is illustrated on financial stock returns.