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χ2χ^2-confidence sets in high-dimensional regression

Abstract

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 1\ell_1-penalty. We show that this procedure leads to an asymptotically χ2\chi^2-distributed pivot, with a remainder term depending only on the 1\ell_1-error of the initial estimator. We show that under 1\ell_1-sparsity conditions on the regression coefficients β0\beta^0 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 β0\beta^0 and compatibility conditions on the design.

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