Uncertainty assessment for generalized linear models via local bootstrapping

Variable selection has been commonly conducted to identify a subset of predictors that significantly contribute to outcomes. Besides obtaining sparse estimates, evaluating estimation uncertainty for parameters is also pivotal. This assessment can be achieved via high-dimensional inference, involving the construction of confidence intervals or hypothesis testing. Different from estimation uncertainty, which reflects the variability in parameter estimates, selection uncertainty reflects the variability in the selection of predictors, including size and composition. Both types of uncertainty can inform the reliability of variable selection procedures. In this paper, we use generalized linear models to either approximate or specify the true data generating process. First, a local residual bootstrap (LRB) is introduced to approximate the distribution of high-dimensional penalized estimates encompassing all values of the regression parameter vector. The simultaneous confidence regions by the LRB can provide insights into estimation uncertainty. Second, by using the LRB, the model confidence bound (MCB) is constructed to evaluate selection uncertainty. Rigorous proofs establish the consistency properties of the LRB and coverage properties of the MCB. To promote implementation, the modified local residual bootstrap (mLRB) is proposed, and two visualization curves are introduced to facilitate the assessment and comparison of selection uncertainty. Simulations demonstrate the proposed approaches' practical utility and superiority over closely related alternatives. In the analysis of two real data sets, the proposed analysis provides valuable insights into the uncertainty of the selection results.