  In this paper we estimate the sparse dependence structure in the tail region
of a multivariate random vector, potentially of high dimension. The tail
dependence is modeled via a graphical model for extremes embedded in the
Huesler-Reiss distribution (Engelke and Hitz, 2020). We propose the extreme
graphical lasso procedure to estimate the sparsity in the tail dependence,
similar to the Gaussian graphical lasso method in high dimensional statistics.
We prove its consistency in identifying the graph structure and estimating
model parameters. The efficiency and accuracy of the proposed method are
illustrated in simulated and real examples.
