  A sparse precision matrix can be directly translated into a sparse Gaussian
graphical model under the assumption that the data follow a joint normal
distribution. This neat property makes high-dimensional precision matrix
estimation very appealing in many applications. However, in practice we often
face nonnormal data, and variable transformation is often used to achieve
normality. In this paper we consider the nonparanormal model that assumes that
the variables follow a joint normal distribution after a set of unknown
monotone transformations. The nonparanormal model is much more flexible than
the normal model while retaining the good interpretability of the latter in
that each zero entry in the sparse precision matrix of the nonparanormal model
corresponds to a pair of conditionally independent variables. In this paper we
show that the nonparanormal graphical model can be efficiently estimated by
using a rank-based estimation scheme which does not require estimating these
unknown transformation functions. In particular, we study the rank-based
graphical lasso, the rank-based neighborhood Dantzig selector and the
rank-based CLIME. We establish their theoretical properties in the setting
where the dimension is nearly exponentially large relative to the sample size.
It is shown that the proposed rank-based estimators work as well as their
oracle counterparts defined with the oracle data. Furthermore, the theory
motivates us to consider the adaptive version of the rank-based neighborhood
Dantzig selector and the rank-based CLIME that are shown to enjoy graphical
model selection consistency without assuming the irrepresentable condition for
the oracle and rank-based graphical lasso. Simulated and real data are used to
demonstrate the finite performance of the rank-based estimators.
