  We consider tests of significance in the setting of the graphical lasso for
inverse covariance matrix estimation. We propose a simple test statistic based
on a subsequence of the knots in the graphical lasso path. We show that this
statistic has an exponential asymptotic null distribution, under the null
hypothesis that the model contains the true connected components.
  Though the null distribution is asymptotic, we show through simulation that
it provides a close approximation to the true distribution at reasonable sample
sizes. Thus the test provides a simple, tractable test for the significance of
new edges as they are introduced into the model. Finally, we show connections
between our results and other results for regularized regression, as well as
extensions of our results to other correlation matrix based methods like
single-linkage clustering.
