Asymptotic Confidence Regions for High-dimensional Structured Sparsity

Abstract

In the setting of high-dimensional linear regression models, we propose two frameworks for constructing pointwise and group confidence sets for penalized estimators which incorporate prior knowledge about the organization of the non-zero coefficients. This is done by desparsifying the estimator as in van de Geer et al. [18] and van de Geer and Stucky [17], then using an appropriate estimator for the precision matrix $\Theta$. In order to estimate the precision matrix a corresponding structured matrix norm penalty has to be introduced. After normalization the result is an asymptotic pivot. The asymptotic behavior is studied and simulations are added to study the differences between the two schemes.

arXiv comment28 pages, 4 figures, 1 table

References in corpus (6)

• High-dimensional graphs and variable selection with the Lasso
• On asymptotically optimal confidence regions and tests for high-dimensional models
• Confidence Intervals and Hypothesis Testing for High-Dimensional Regression
• SLOPE - Adaptive variable selection via convex optimization
• Structured Sparsity and Generalization
• Convex Relaxation for Combinatorial Penalties

Cited by in corpus (3)

• Joint Estimation and Inference for Data Integration Problems based on Multiple Multi-layered Gaussian Graphical Models
• Statistical control for spatio-temporal MEG/EEG source imaging with desparsified multi-task Lasso
• Chi-square and normal inference in high-dimensional multi-task regression