Detection of Block-Exchangeable Structure in Large-Scale Correlation Matrices
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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; an assumption often made in applications 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. Even in the unstructured case K = d, though there is no gain asymptotically, the new estimator can be much more efficient in finite samples. When the data are elliptical, the results extend to linear correlation matrices and their inverses. The procedure is illustrated on financial stock returns.
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