## The Annals of Statistics

### High-dimensional semiparametric Gaussian copula graphical models

#### Abstract

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: http://cran.r-project.org/.

#### Article information

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

Dates
First available in Project Euclid: 23 January 2013

http://projecteuclid.org/euclid.aos/1358951383

Digital Object Identifier
doi:10.1214/12-AOS1037

Mathematical Reviews number (MathSciNet)
MR3059084

Zentralblatt MATH identifier
1297.62073

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

#### Citation

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. http://projecteuclid.org/euclid.aos/1358951383.

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