  In this paper, we propose a semiparametric approach, named nonparanormal
skeptic, for efficiently and robustly estimating high dimensional undirected
graphical models. To achieve modeling flexibility, we consider Gaussian Copula
graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To
achieve estimation robustness, we exploit nonparametric rank-based correlation
coefficient estimators, including Spearman's rho and Kendall's tau. In high
dimensional settings, we prove that the nonparanormal skeptic achieves the
optimal parametric rate of convergence in both graph and parameter estimation.
This celebrating result suggests that the Gaussian copula graphical models can
be used as a safe replacement of the popular Gaussian graphical models, even
when the data are truly Gaussian. Besides theoretical analysis, we also conduct
thorough numerical simulations to compare different estimators for their graph
recovery performance under both ideal and noisy settings. The proposed methods
are then applied on a large-scale genomic dataset to illustrate their empirical
usefulness. The R language software package huge implementing the proposed
methods is available on the Comprehensive R Archive Network: http://cran.
r-project.org/.
