  We study confidence regions and approximate chi-squared tests for variable
groups in high-dimensional linear regression. When the size of the group is
small, low-dimensional projection estimators for individual coefficients can be
directly used to construct efficient confidence regions and p-values for the
group. However, the existing analyses of low-dimensional projection estimators
do not directly carry through for chi-squared-based inference of a large group
of variables without inflating the sample size by a factor of the group size.
We propose to de-bias a scaled group Lasso for chi-squared-based statistical
inference for potentially very large groups of variables. We prove that the
proposed methods capture the benefit of group sparsity under proper conditions,
for statistical inference of the noise level and variable groups, large and
small. Such benefit is especially strong when the group size is large.
