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

Abstract

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.

Published in at http://dx.doi.org/10.1214/12-AOS1041 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (7)

• High-dimensional graphs and variable selection with the Lasso
• Simultaneous analysis of Lasso and Dantzig selector
• Sparse permutation invariant covariance estimation
• Sparsistency and rates of convergence in large covariance matrix estimation
• Regularized rank-based estimation of high-dimensional nonparanormal graphical models
• Multiple Testing and Error Control in Gaussian Graphical Model Selection
• High-dimensional covariance estimation based on Gaussian graphical models

Cited by in corpus (14)

• Challenges of Big Data Analysis
• Regularized rank-based estimation of high-dimensional nonparanormal graphical models
• Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses
• Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix
• Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution
• Confidence regions for entries of a large precision matrix
• Adaptive estimation of the copula correlation matrix for semiparametric elliptical copulas
• Introduction to correlation networks: Interdisciplinary approaches beyond thresholding
• Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation
• Limiting spectral distribution of large dimensional Spearman's rank correlation matrices
• Detection of Block-Exchangeable Structure in Large-Scale Correlation Matrices
• latentcor: An R Package for estimating latent correlations from mixed data types
• A review of Gaussian Markov models for conditional independence
• On generating random Gaussian graphical models