The Annals of Statistics

High-dimensional semiparametric Gaussian copula graphical models

Han Liu, Fang Han, Ming Yuan, John Lafferty, and Larry Wasserman

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We propose a semiparametric approach called the nonparanormal SKEPTIC for efficiently and robustly estimating high-dimensional undirected graphical models. To achieve modeling flexibility, we consider the nonparanormal graphical models proposed by Liu, Lafferty and Wasserman [J. Mach. Learn. Res. 10 (2009) 2295–2328]. To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman’s rho and Kendall’s tau. We prove that the nonparanormal SKEPTIC achieves the optimal parametric rates of convergence for both graph recovery and parameter estimation. This result suggests that the nonparanormal graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare the graph recovery performance of different estimators under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic data set to illustrate their empirical usefulness. The R package huge implementing the proposed methods is available on the Comprehensive R Archive Network:

Article information

Ann. Statist. Volume 40, Number 4 (2012), 2293-2326.

First available in Project Euclid: 23 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 62F12: Asymptotic properties of estimators

High-dimensional statistics undirected graphical models Gaussian copula nonparanormal graphical models robust statistics minimax optimality biological regulatory networks


Liu, Han; Han, Fang; Yuan, Ming; Lafferty, John; Wasserman, Larry. High-dimensional semiparametric Gaussian copula graphical models. Ann. Statist. 40 (2012), no. 4, 2293--2326. doi:10.1214/12-AOS1037.

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