Weak signal identification and inference in penalized model selection

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. This thesis propose an identification procedure for weak signals in finite samples, and provide a transition phase in-between noise and strong signal strengths. A new two-step inferential method is introduced to construct better confidence intervals for the identified weak signals. 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 outperforms the perturbation and bootstrap resampling approaches. The method is illustrated for HIV antiretroviral drug susceptibility data to identify genetic mutations associated with HIV drug resistance. We also provide signal's inference method based on the exact distribution of penalized estimator. The finite sample distribution is quite different from its asymptotic counterpart, which can be highly non-normal with a point mass at zero. Numerical studies indicate that the density-based approach works well when true parameter is moderately large. However, it cannot provide accurate inference when signal is weak