  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 random variables but also
critical for improving statistical estimation precision. In high-dimensional
data analysis, despite an abundant literature on estimating this graph
structure, tests for the adequacy of its specification at a global level is
severely underdeveloped. To make progress, this paper proposes a novel
goodness-of-fit test that is computationally easy and theoretically tractable.
Under the null hypothesis, it is shown that asymptotic distribution of the
proposed test statistic follows a Gumbel distribution. Interestingly the
location parameter of this limiting Gumbel distribution depends on the
dependence structure under the null. We further develop a novel
consistency-empowered test statistic when the true structure is nested in the
postulated structure, by amplifying the noise incurred in estimation. Extensive
simulation illustrates that the proposed test procedure has the right size
under the null, and is powerful under the alternative. As an application, we
apply the test 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.
