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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.

The goal of this study was to test the equality of two covariance matrices by using modified Pillai's trace statistics under a high-dimensional framework, i.e., the dimension and sample sizes go to infinity proportionally. In this paper, we introduce two modified Pillai's trace statistics and obtain their asymptotic distributions under the null hypothesis. The benefits of the proposed statistics include the following: (1) the sample size can be smaller than the dimensions; (2) the limiting distributions of the proposed statistics are universal; and (3) we do not restrict the structure of the population covariance matrices. The theoretical results are established under mild and practical assumptions, and their properties are demonstrated numerically by simulations and a real data analysis., Comment: 31 pages-to appear in Journal of Statistical Planning and Inference

This paper considers the optimal modification of the likelihood ratio test (LRT) for the equality of two high-dimensional covariance matrices. The classical LRT is not well defined when the dimensions are larger than or equal to one of the sample sizes. In this paper, an optimally modified test that works well in cases where the dimensions may be larger than the sample sizes is proposed. In addition, the test is established under the weakest conditions on the moments and the dimensions of the samples. We also present weakly consistent estimators of the fourth moments, which are necessary for the proposed test, when they are not equal to 3. From the simulation results and real data analysis, we find that the performances of the proposed statistics are robust against affine transformations.