Asymptotic Confidence Regions for High-Dimensional Structured Sparsity

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 nonzero coefficients. This is done by desparsifying the estimator by S. van de Geer and B. Stucky and S. van de Geer et al., then using an appropriate estimator for the precision matrix Θ. 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.