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

By Lingzhou Xue, Hui Zou. 2012

274 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Statistics and Probability

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Citations in the corpus

Kernel‐based tests for joint independence (2016, background): “…correlations (Székely and Rizzo, 2009, 2014), rank-based correlations (Bergsma and Dassios, 2014; Leung and Drton, 2016; Nandy et al., 2016) or also non-parametric and semiparametric copula-based correlations (Liu et al., 2012; Xue and Zou, 2012; Wegkamp and Zhao, 2016; Gaißer et al., 2010).”
Sufficient Forecasting Using Factor Models (2015, methodology): “For example, Forni et al. (2000, 2015) studied factor estimation based on projection, Liu et al. (2012), Xue and Zou (2012) and Fan et al. (2015b) employed rank-based methods for robust estimation of the covariance matrices, and Connor et al. (2012) applied a weighted additive nonparametric…”
Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation (2012, methodology): “Xue and Zou (2012) estimated the nonparanormal model by applying CLIME (and graphical lasso, neighborhood Dantzig selector) to the adjusted Spearman’s rank correlations.”
PROJECTED PRINCIPAL COMPONENT ANALYSIS IN FACTOR MODELS (2014, methodology): “These have further expanded into robust estimation based on Kendall’s tau estimation (Liu et al., 2012; Xue and Zou, 2012).”
High‐dimensional covariance matrix estimation (2019, methodology): “Xue and Zou (2012) proposed to use the nonparanormal model (4) for x.”
Multitask Quantile Regression under the Transnormal Model (2016, methodology): “Models 1–3 are autoregressive models considered by Huang et al. (2006) and Xue and Zou (2012).”
Challenges of Big Data Analysis (2013, unknown): “However,ithasa greaterbalancebetweendiseasedandnon-diseased subjects.”
Multi-task Sparse Structure Learning with Gaussian Copula Models (2016, background): “A more general formulation of the Gaussian copula is the semiparametric Gaussian copulas (Liu et al., 2009; Xue and Zou, 2012), which allows the marginals to follow any non-parametric distribution.”
Structure Learning in Graphical Modeling (2016, background): “Liu et al. (2012a) and Xue & Zou (2012) observe that eﬃcient estimation in the copula models can be based on Kendall’s τ and Spearman’s ρ .”
Analysis of elliptical copula correlation factor model with Kendall's tau (2013, methodology): “Inspired by this observation, both [28] and [46] employ Σ̂, a variant of Σ̂ using Spearman’s rho statistic, to study the estimation of precision matrix under this setting.”
Efficient Estimation of Approximate Factor Models (2012, methodology): “The additive δT was not included in the original definition of adaptive lasso in Zou (2006), but has often been seen in recent literature, e.”
An Overview on the Estimation of Large Covariance and Precision Matrices (2015, unknown): “The literature has been further expanded into robust estimation based on regularized rank-based approaches (Liu et al., 2012a; Xue and Zou, 2012).”