  We study the problem of identifying the set of \emph{active} variables,
termed in the literature as \emph{variable selection} or \emph{multiple
hypothesis testing}, depending on the pursued criteria. For a general
\emph{robust setting} of non-normal, possibly dependent observations and a
generalized notion of \emph{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 \emph{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 \emph{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 \emph{uncertainty quantification problem} in the variable
selection and multiple testing context in our robust setting.
