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 rankbased 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. Received February 2012; revised August 2012. 1 Some results in this paper were reported in a research proposal funded by the Office of Naval Research in November 2010. Supported in part by NSF Grant DMS-08-46068 and a grant from ONR. AMS 2000 subject classifications. Primary 62G05, 62G20; secondary 62F12, 62J07. Key words and phrases. CLIME, Dantzig selector, graphical lasso, nonparanormal graphical model, rate of convergence, variable transformation. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2012, Vol. 40, No. 5, 2541-2571. This reprint differs from the original in pagination and typographic detail. 1 2 L. XUE AND H. ZOU 1. Introduction. Estimating covariance or precision matrices is of fundamental importance in multivariate statistical methodologies and applications. In particular, when data follow a joint normal distribution, X = (X 1 , . . . , X p ) ∼ N p (µ, Σ), the precision matrix Θ = Σ −1 can be directly translated into a Gaussian graphical model. The Gaussian graphical model serves as a noncausal structured approach to explore the complex systems consisting of Gaussian random variables, and it finds many interesting applications in areas such as gene expression genomics and macroeconomics determinants study [Friedman (2004) , Wille et al. (2004) , Dobra, Eicher and Lenkoski (2010)]. The precision matrix plays a critical role in the Gaussian graphical models because the zero entries in Θ = (θ ij ) p×p precisely capture the desired conditional independencies, that is, θ ij = 0 if and only if Lauritzen (1996) , Edwards (2000)]. The sparsity pursuit in precision matrices was initially considered by Dempster (1972) as the covariance selection problem. Multiple testing methods have been employed for network exploration in the Gaussian graphical models [Drton and Perlman (2004) ]. With rapid advances of the highthroughput technology (e.g., microarray, functional MRI), estimation of a sparse graphical model has become increasingly important in the highdimensional setting. Some well-developed penalization techniques have been used for estimating sparse Gaussian graphical models. In a highly-cited paper, Meinshausen and Bühlmann (2006) proposed the neighborhood selection scheme which tries to discover the smallest index set ne α for each variable X α satisfying X α ⊥ ⊥ X \ {X α , X neα }|X neα . Meinshausen and Bühlmann (2006) further proposed to use the lasso [Tibshirani (1996) ] to fit each neighborhood regression model. Afterwards, one can summarize the zero patterns by aggregation via union or intersection. Yuan (2010) considered the Dantzig selector [Candes and Tao (2007) ] as an alternative to the lasso penalized least squares in the neighborhood selection scheme. Peng et al. (2009) [Fan and Li (2001) ] is used to replace the lasso penalty in order to overcome the bias issue of the lasso penalization. Zhou et al. (2011) proposed a hybrid method for estimating sparse Gaussian graphical models: they first infer a sparse Gaussian graphical model structure via thresholding neighborhood selection and then estimate the precision
In two recent papers REF , the rank-based estimators were used to estimate correlation matrixŜ and then fit the following model GGMs.
Regularized rank-based estimation of high-dimensional nonparanormal graphical models
Annals of Statistics 2012, Vol. 40, No. 5, 2541-2571
Mathematics