  Simultaneous inference after model selection is of critical importance to
address scientific hypotheses involving a set of parameters. In this paper, we
consider high-dimensional linear regression model in which a regularization
procedure such as LASSO is applied to yield a sparse model. To establish a
simultaneous post-model selection inference, we propose a method of contraction
and expansion (MOCE) along the line of debiasing estimation that enables us to
balance the bias-and-variance trade-off so that the super-sparsity assumption
may be relaxed. We establish key theoretical results for the proposed MOCE
procedure from which the expanded model can be selected with theoretical
guarantees and simultaneous confidence regions can be constructed by the joint
asymptotic normal distribution. In comparison with existing methods, our
proposed method exhibits stable and reliable coverage at a nominal significance
level with substantially less computational burden, and thus it is trustworthy
for its application in solving real-world problems.
