Implementable confidence sets in high dimensional regression
This addresses the challenge of uncertainty quantification in high-dimensional statistics for researchers and practitioners, though it appears incremental as it builds on existing confidence set frameworks.
The paper tackles the problem of constructing adaptive and honest confidence sets for sparse parameters in high-dimensional linear regression, where the sparsity level is unknown, and presents a method that is implementable in practice.
We consider the setting of linear regression in high dimension. We focus on the problem of constructing adaptive and honest confidence sets for the sparse parameter θ, i.e. we want to construct a confidence set for theta that contains theta with high probability, and that is as small as possible. The l_2 diameter of a such confidence set should depend on the sparsity S of θ- the larger S, the wider the confidence set. However, in practice, S is unknown. This paper focuses on constructing a confidence set for θwhich contains θwith high probability, whose diameter is adaptive to the unknown sparsity S, and which is implementable in practice.