Large multiple graphical model inference via bootstrap

Yongli Zhang, Xiaotong Shen, Shaoli Wang

Research output: Contribution to journalArticlepeer-review


Large economic and financial networks may experience stage-wise changes as a result of external shocks. To detect and infer a structural change, we consider an inference problem within a framework of multiple Gaussian Graphical Models when the number of graphs and the dimension of graphs increase with the sample size. In this setting, two major challenges emerge as a result of the bias and uncertainty inherent in the regularization required to treat such overparameterized models. To deal with these challenges, the bootstrap method is utilized to approximate the sampling distribution of a likelihood ratio test statistic. We show theoretically that the proposed method leads to a correct asymptotic inference in a high-dimensional setting, regardless of the distribution of the test statistic. Simulations show that the proposed method compares favorably to its competitors such as the Likelihood Ratio Test. Finally, our statistical analysis of a network of 200 stocks reveals that the interacting units in the financial network become more connected as a result of the financial crisis between 2007 and 2009. More importantly, certain units respond more strongly than others. Furthermore, after the crisis, some changes weaken, while others strengthen.

Original languageEnglish (US)
Pages (from-to)695-717
Number of pages23
JournalStatistica Sinica
Issue number2
StatePublished - Apr 2020

Bibliographical note

Funding Information:
This research was supported in part by NSF grants DMS-1415500, DMS-1712564, and DMS-1721216, and NIH grants 1R01GM081535-01, HL65462, and R01HL105397. Wang’s research was partially supported by the National Natural Science Foundation of China NSFC, 11371235. The authors would like to thank the editor, associate editor, and anonymous referees for their helpful comments and suggestions.


  • Bootstrap
  • Graphical models
  • High-dimensional inference
  • Model selection
  • Regularization

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