  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.
