We study a high-dimensional regression model. Aim is to construct a confidence set for a given group of regression coefficients, treating all other regression coefficients as nuisance parameters. We apply a one-step procedure with the square-root Lasso as initial estimator and a multivariate square-root Lasso for constructing a surrogate Fisher information matrix. The multivariate square-root Lasso is based on nuclear norm loss with l1-penalty. We show that this procedure leads to an asymptotically χ~2-distributed pivot, with a remainder term depending only on the l1-error of the initial estimator. We show that under l1-sparsity conditions on the regression coefficients /8° the square-root Lasso produces to a consistent estimator of the noise variance and we establish sharp oracle inequalities which show that the remainder term is small under further sparsity conditions on β~0 and compatibility conditions on the design.
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