Open Access
October 2012 Regularized rank-based estimation of high-dimensional nonparanormal graphical models
Lingzhou Xue, Hui Zou
Ann. Statist. 40(5): 2541-2571 (October 2012). DOI: 10.1214/12-AOS1041


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.


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Lingzhou Xue. Hui Zou. "Regularized rank-based estimation of high-dimensional nonparanormal graphical models." Ann. Statist. 40 (5) 2541 - 2571, October 2012.


Published: October 2012
First available in Project Euclid: 4 February 2013

zbMATH: 1373.62138
MathSciNet: MR3097612
Digital Object Identifier: 10.1214/12-AOS1041

Primary: 62G05 , 62G20
Secondary: 62F12 , 62J07

Keywords: CLIME , Dantzig selector , graphical lasso , nonparanormal graphical model , rate of convergence , variable transformation

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 5 • October 2012
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