Uncertainty quantification for robust variable selection and multiple testing
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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.
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