The Annals of Statistics

Regularized rank-based estimation of high-dimensional nonparanormal graphical models

Lingzhou Xue and Hui Zou

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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.

Article information

Ann. Statist., Volume 40, Number 5 (2012), 2541-2571.

First available in Project Euclid: 4 February 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 62F12: Asymptotic properties of estimators 62J07: Ridge regression; shrinkage estimators

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


Xue, Lingzhou; Zou, Hui. Regularized rank-based estimation of high-dimensional nonparanormal graphical models. Ann. Statist. 40 (2012), no. 5, 2541--2571. doi:10.1214/12-AOS1041.

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Supplemental materials

  • Supplementary material: Supplement material for “Regularized rank-based estimation of high-dimensional nonparanormal graphical models”. In this supplementary note, we give the complete proofs of Theorems 2 and 5.