Uncertainty quantification for robust variable selection and multiple testing

Abstract

We study the problem of identifying the set of active variables, termed in the literature as variable selection or multiple hypothesis testing, depending on the pursued criteria. For a general robust setting of non-normal, possibly dependent observations and a generalized notion of active set, we propose a procedure that is used simultaneously for the both tasks, variable selection and multiple testing. The procedure is based on the risk hull minimization method, but can also be obtained as a result of an empirical Bayes approach or a penalization strategy. We address its quality via various criteria: the Hamming risk, FDR, FPR, FWER, NDR, FNR, and various multiple testing risks, e.g., MTR=FDR+NDR; and discuss a weak optimality of our results. Finally, we introduce and study, for the first time, the uncertainty quantification problem in the variable selection and multiple testing context in our robust setting.