Inferences on blockwise correlation matrix with unknown group structure

The covariance matrix of a blockwise correlation matrix has become increasingly prevalent in high-dimensional data analysis. However, accurate estimation of such covariance matrices poses significant challenges. One major difficulty is that the maximum likelihood estimation fails to ensure positive semi-definiteness. Moreover, the block number and group memberships of variables are typically unknown in practice. To address these challenges, a novel two-stage blockwise correlation matirx estimation method is proposed to estimate the covariance matrix of p variables with an unknown blockwise correlation matrix structure. In the first stage, the number of blocks and group memberships are estimated using the ridge-type ratio criterion and spectral clustering respectively, and consistency of these estimators is established. In the second stage, a closed-form estimator for the blockwise correlation matrix is developed by utilizing the sample moments of variable means within estimated groups, which subsequently yields the covariance matrix estimator. By appropriately controlling the estimation error in group memberships, theoretical analysis shows the asymptotic normality of the correlation coefficient estimators and the stochastic convergence rates of the estimated blockwise correlation matrix and corresponding estimated covariance matrix under certain moment conditions. Extensive simulations and an empirical study of stock returns in the Chinese stock market are analyzed to illustrate the usefulness of the proposed methods.