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Asymptotic Confidence Regions for High-Dimensional Structured Sparsity

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 matrixΘ. 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.


A weakly decomposable norm is in some sense able to split up into two norms, one norm measuring the size of the vector on the active set and the other norm the size on its complement. The weakly decomposable norm itself reflects the prior information of the underlying sparsity.

The estimation of the precision matrix can be done in two ways which are beneficial for the construction of asymptotic confidence regions. These two frameworks differ in the structure of the penalty function. The theoretical behavior and the assumptions on the sparsity is studied.


Two frameworks for penalized estimators which incorporate structured sparsity patterns have been proposed.

The first framework makes use of the gauge function, which is in most cases an type norm due to the additivity of the lower bounding weak decomposable norms. The second framework is penalized by the structured sparse norm itself.

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