The modified Cholesky decomposition is popular for inverse covariance estimation, but often needs pre-specification on the full information of variable ordering. In this work, we propose a block Cholesky decomposition (BCD) for estimating inverse covariance matrix under the partial information of variable ordering, in the sense that the variables can be divided into several groups with available ordering among groups, but variables within each group have no orderings. The proposed BCD model provides a unified framework for several existing methods including the modified Cholesky decomposition and the Graphical lasso. By utilizing the partial information on variable ordering, the proposed BCD model guarantees the positive definiteness of the estimated matrix with statistically meaningful interpretation. Theoretical results are established under regularity conditions. Simulation and case studies are conducted to evaluate the proposed BCD model.

翻译：修正乔列斯基分解在逆协方差估计中应用广泛，但通常需要预先指定变量排序的全部信息。本文提出一种块状乔列斯基分解（BCD）方法，用于在变量排序部分信息条件下估计逆协方差矩阵——即变量可划分为若干组，组间排序已知，但组内变量无排序。所提出的BCD模型为包括修正乔列斯基分解和图形套索在内的多种现有方法提供了统一框架。通过利用变量排序的部分信息，该模型能保证估计矩阵的正定性，并具有统计学意义的可解释性。在正则性条件下建立了理论结果，并通过仿真和案例研究评估了所提出的BCD模型。