Uncertainty Quantification Under Group Sparsity

Quantifying the uncertainty in a penalized estimator under group sparsity, such as the group Lasso, is an important, yet still open, question. We establish, under a high-dimensional scaling, the consistency of an estimated sampling distribution for the group Lasso, assuming a normal error model and mild conditions on the design matrix and the true coefficients. Consequently, simulation from the estimated sampling distribution provides a valid and convenient means of constructing interval estimates for both individual coefficients and potentially large groups of coefficients. The results are further generalized to other group norm penalties and sub-Gaussian errors.