Local and Global Inference for High Dimensional Gaussian Copula Graphical Models

In this paper, we propose a unified asymptotic inference framework for Gaussian copula graphical models, to test the presence of a single edge and construct a uniform confidence subgraph for the true graph. To avoid the estimation of the unknown marginal transformation functions, we consider a pseudo likelihood approach. The core of our framework is a decorrelated pseudo score function for the parameter of interest. The proposed pseudo score function has the structure of a nonlinear transformation of U-statistics, and a delta method-type analysis is used to prove its asymptotic normality. Our theory provides asymptotic guarantees on the type I error as well as the local power of the proposed score test. We also propose a global hypothesis testing procedure based on Wald test statistic and multiplier bootstrap, to provide a confidence subgraph for the true graph. The theoretical properties of the proposed inferential methods are verified by thorough numerical experiments.

[1]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[2]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[3]  C. Klaassen,et al.  Efficient estimation in the bivariate normal copula model: normal margins are least favourable , 1997 .

[4]  A. V. D. Vaart,et al.  Asymptotic Statistics: U -Statistics , 1998 .

[5]  Wenjiang J. Fu,et al.  Asymptotics for lasso-type estimators , 2000 .

[6]  Mathias Drton,et al.  A SINful approach to Gaussian graphical model selection , 2005 .

[7]  N. Meinshausen,et al.  High-dimensional graphs and variable selection with the Lasso , 2006, math/0608017.

[8]  M. Yuan,et al.  Model selection and estimation in the Gaussian graphical model , 2007 .

[9]  D. Koller,et al.  Population genomics of human gene expression , 2007, Nature Genetics.

[10]  M. Drton,et al.  Multiple Testing and Error Control in Gaussian Graphical Model Selection , 2005, math/0508267.

[11]  Alexandre d'Aspremont,et al.  Model Selection Through Sparse Maximum Likelihood Estimation , 2007, ArXiv.

[12]  R. Tibshirani,et al.  Sparse inverse covariance estimation with the graphical lasso. , 2008, Biostatistics.

[13]  Liang Chen,et al.  Considering dependence among genes and markers for false discovery control in eQTL mapping , 2008, Bioinform..

[14]  Bin Yu,et al.  High-dimensional covariance estimation by minimizing ℓ1-penalized log-determinant divergence , 2008, 0811.3628.

[15]  Adam J. Rothman,et al.  Sparse permutation invariant covariance estimation , 2008, 0801.4837.

[16]  Pei Wang,et al.  Partial Correlation Estimation by Joint Sparse Regression Models , 2008, Journal of the American Statistical Association.

[17]  Larry A. Wasserman,et al.  The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs , 2009, J. Mach. Learn. Res..

[18]  Jianqing Fan,et al.  Sparsistency and Rates of Convergence in Large Covariance Matrix Estimation. , 2007, Annals of statistics.

[19]  Larry A. Wasserman,et al.  Stability Approach to Regularization Selection (StARS) for High Dimensional Graphical Models , 2010, NIPS.

[20]  Ming Yuan,et al.  High Dimensional Inverse Covariance Matrix Estimation via Linear Programming , 2010, J. Mach. Learn. Res..

[21]  Cun-Hui Zhang,et al.  Confidence intervals for low dimensional parameters in high dimensional linear models , 2011, 1110.2563.

[22]  Cun-Hui Zhang,et al.  Scaled sparse linear regression , 2011, 1104.4595.

[23]  A. Belloni,et al.  Inference for High-Dimensional Sparse Econometric Models , 2011, 1201.0220.

[24]  T. Cai,et al.  A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation , 2011, 1102.2233.

[25]  Ali Jalali,et al.  High-dimensional Sparse Inverse Covariance Estimation using Greedy Methods , 2011, AISTATS.

[26]  A. Mohammadi,et al.  Bayesian Structure Learning in Sparse Gaussian Graphical Models , 2012, 1210.5371.

[27]  H. Zou,et al.  Regularized rank-based estimation of high-dimensional nonparanormal graphical models , 2012, 1302.3082.

[28]  Xiaotong Shen,et al.  Journal of the American Statistical Association Likelihood-based Selection and Sharp Parameter Estimation Likelihood-based Selection and Sharp Parameter Estimation , 2022 .

[29]  Larry A. Wasserman,et al.  High Dimensional Semiparametric Gaussian Copula Graphical Models. , 2012, ICML 2012.

[30]  Po-Ling Loh,et al.  Regularized M-estimators with nonconvexity: statistical and algorithmic theory for local optima , 2013, J. Mach. Learn. Res..

[31]  Larry A. Wasserman,et al.  Estimating Undirected Graphs Under Weak Assumptions , 2013, ArXiv.

[32]  Xiaotong Shen,et al.  Simultaneous Grouping Pursuit and Feature Selection Over an Undirected Graph , 2013, Journal of the American Statistical Association.

[33]  Kengo Kato,et al.  Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors , 2013 .

[34]  P. Bühlmann.  Statistical significance in high-dimensional linear models , 2013 .

[35]  Kengo Kato,et al.  Comparison and anti-concentration bounds for maxima of Gaussian random vectors , 2013, 1301.4807.

[36]  Cun-Hui Zhang,et al.  Sparse matrix inversion with scaled Lasso , 2012, J. Mach. Learn. Res..

[37]  Bani K Mallick,et al.  Joint High‐Dimensional Bayesian Variable and Covariance Selection with an Application to eQTL Analysis , 2013, Biometrics.

[38]  Weidong Liu.  Gaussian graphical model estimation with false discovery rate control , 2013, 1306.0976.

[39]  Alessandro Rinaldo,et al.  Berry-Esseen bounds for estimating undirected graphs , 2014 .

[40]  S. Geer,et al.  On asymptotically optimal confidence regions and tests for high-dimensional models , 2013, 1303.0518.

[41]  Adel Javanmard,et al.  Confidence intervals and hypothesis testing for high-dimensional regression , 2013, J. Mach. Learn. Res..

[42]  R. Tibshirani,et al.  A SIGNIFICANCE TEST FOR THE LASSO. , 2013, Annals of statistics.

[43]  R. Tibshirani,et al.  Exact Post-selection Inference for Forward Stepwise and Least Angle Regression , 2014 .

[44]  H. Zou,et al.  High dimensional semiparametric latent graphical model for mixed data , 2014, 1404.7236.

[45]  S. Geer,et al.  Confidence intervals for high-dimensional inverse covariance estimation , 2014, 1403.6752.

[46]  Xiaotong Shen,et al.  Structural Pursuit Over Multiple Undirected Graphs , 2014, Journal of the American Statistical Association.

[47]  Peter Wonka,et al.  Fused Multiple Graphical Lasso , 2012, SIAM J. Optim..

[48]  Harrison H. Zhou,et al.  Asymptotic normality and optimalities in estimation of large Gaussian graphical models , 2013, 1309.6024.