  Weak signal identification and inference are very important in the area of
penalized model selection, yet they are under-developed and not well-studied.
Existing inference procedures for penalized estimators are mainly focused on
strong signals. In this paper, we propose an identification procedure for weak
signals in finite samples, and pro- vide a transition phase in-between noise
and strong signal strengths. We also introduce a new two-step inferential
method to construct better confidence intervals for the identified weak
signals. Our theory development assumes that variables are orthogonally
designed. Both theory and numerical studies indicate that the proposed method
leads to better confidence coverage for weak signals, compared with those using
asymptotic inference. In addition, the proposed method out- performs the
perturbation and bootstrap resampling approaches. We illustrate our method for
HIV antiretroviral drug susceptibility data to identify genetic mutations
associated with HIV drug resistance.
