Abstract

The Gaussian graphical model is routinely employed to model the joint distribution of multiple random variables. The graph it induces is not only useful for

describing the relationship between these variables but also critical for improving statistical estimation precision. In high-dimensional data analysis, despite

abundant literature on estimating this graph structure, tests for the adequacy of

its specification at a global level are severely underdeveloped. To make progress,

this paper proposes novel goodness-of-fit tests that are computationally easy and

theoretically tractable. The first contribution of this paper is the development of

a new direct plug-in test statistic. We show that its asymptotic distribution under

the null follows a Gumbel distribution with a location parameter depending on

the underlying true graph structure. The direct test, however, has no power for

detecting structures including the truth but not equal. Our second contribution is

the development of a novel consistency-empowered test statistic that gains power

by, interestingly, amplifying the noise incurred in estimation. The improved test is

shown to be universally consistent for all fixed alternatives. Extensive simulation

illustrates that the proposed test procedures have the right size under the null,

and is powerful under alternatives. As an application, we apply the tests to the

analysis of a COVID-19 data set, demonstrating that our test can serve as a

valuable tool in choosing a graph structure to improve estimation efficiency.

Information

Preprint No.SS-2023-0385
Manuscript IDSS-2023-0385
Complete AuthorsThien-Minh Le, Ping-Shou Zhong, Chenlei Leng
Corresponding AuthorsPing-Shou Zhong
Emailspszhong@uic.edu

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Acknowledgments

The research was partially supported by NSF grants DMS-1462156 and FRG-

  1. The authors thank Editor Prof. Judy Wang, the Associate Editor,

and the two referees for their constructive comments, which significantly

improved the paper.

Supplementary Materials

All technical proofs, examples of structures satisfying condition (C1),

additional simulation results, a data-driven tuning parameter selection

procedure, and further details on the real data analysis are provided

in the supplementary materials.

The source codes and real data sets

are included in a Github link: https://github.com/leminhthien2011/

UniversalConsistentTestforGraphsofGGM.

Supplementary materials are available for download.