Detection of Block-Exchangeable Structure in Large-Scale Correlation Matrices
This addresses the challenge of uncovering hidden dependence patterns in large-scale multivariate data for fields like finance and econometrics, offering an incremental improvement in estimation efficiency.
The paper tackles the problem of noisy correlation matrix estimation in high-dimensional data by proposing a robust algorithm to detect block-exchangeable structures, which reduces parameters from O(d^2) to O(K^2) and improves performance over sample Kendall rank correlation matrices, with gains shown in finite samples even when K = d.
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.