# Application of fused graphical lasso to statistical inference for multiple sparse precision matrices

## TL;DR

This paper introduces a statistical method to estimate and test relationships in data from multiple groups, using a technique called fused graphical lasso.

## Contribution

The novel contribution is extending fused graphical lasso for statistical inference across multiple sparse precision matrices with a de-biasing technique.

## Key findings

- The fused graphical lasso estimator satisfies an oracle inequality in high-dimensional settings.
- A de-biasing method enables hypothesis testing for linear combinations of precision matrix entries across groups.
- Simulation and real data applications show the method performs well in high-dimensional scenarios.

## Abstract

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. We apply a de-biasing technology, which is used to obtain a new consistent estimator with known distribution for implementing the statistical inference, and extend the statistical inference problem to multiple populations. The corresponding de-biasing FGL estimator and its asymptotic theory are provided. A simulation study and an application of the diffuse large B-cell lymphoma data show that the proposed test works well in high-dimensional situation.