In this paper, the fused graphical lasso (FGL) method is used to estimate multiple precision matrices from multiple populations simultaneously. The lasso penalty in the FGL model is a restraint on sparsity of precision matrices, and a moderate penalty on the two precision matrices from distinct groups restrains the similar structure across multiple groups. In high-dimensional settings, an oracle inequality is provided for FGL estimators, which is necessary to establish the central limit law. We not only focus on point estimation of a precision matrix, but also work on hypothesis testing for a linear combination of the entries of multiple precision matrices. Inspired by Jankova a and van de Geer [confidence intervals for high-dimensional inverse covariance estimation, Electron. J. Stat. 9(1) (2015) 1205-1229.], who investigated a de-biasing technology to obtain a new consistent estimator with known distribution for implementing the statistical inference, we extend the statistical inference problem to multiple populations, and propose the de-biasing FGL estimators. The corresponding asymptotic property of de-biasing FGL estimators is provided. A simulation study shows that the proposed test works well in high-dimensional situations.

翻译：本文采用融合图形套索(FGL)方法同时估计多个总体的精度矩阵。FGL模型中的套索惩罚对精度矩阵施加稀疏性约束，而对来自不同组别的两个精度矩阵施加适度惩罚则约束了多组间的相似结构。在高维设定下，我们给出了FGL估计量的奥拉奇不等式，该不等式是建立中心极限定理的必要条件。本文不仅关注精度矩阵的点估计，还针对多个精度矩阵元素的线性组合进行假设检验。受Jankova与van de Geer[高维逆协方差估计的置信区间，Electron. J. Stat. 9(1) (2015) 1205-1229]启发（该研究提出了一种去偏技术以获得具有已知分布的一致估计量以实现统计推断），我们将统计推断问题推广至多个总体，并提出了去偏FGL估计量。本文给出了去偏FGL估计量的渐近性质，模拟研究表明所提出的检验方法在高维情形下表现良好。