Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $\chi^2$-confidence sets in high-dimensional regression   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 $\ell_1$-penalty. We show
that this procedure leads to an asymptotically $\chi^2$-distributed pivot, with
a remainder term depending only on the $\ell_1$-error of the initial estimator.
We show that under $\ell_1$-sparsity conditions on the regression coefficients
$\beta^0$ 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 $\beta^0$ and
compatibility conditions on the design.

33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph