Covariance estimation with uniform blocks

Estimating a covariance matrix is central to high-dimensional data analysis. The proposed method is motivated by the dependence pattern analyses of multiple types of high-dimensional biomedical data including but not limited to genomics, proteomics, microbiome, and neuroimaging data. The correlation matrices of these biomedical data all demonstrate a well-organized block pattern. In this pattern, the positive and negative pair-wise correlations with large absolute values, are mainly concentrated within diagonal and off-diagonal blocks. We develop a covariance- and precision-matrix estimation framework to fully leverage the organized block pattern. We propose new best unbiased covariance- and precision-matrix estimators in closed forms, and develop theories for the asymptotic proprieties of estimators in both scenarios where the number of blocks is less or greater than the sample size. The simulation and data example analyses show that our method is robust and improves the accuracy of covariance- and precision-matrix estimation.