  In this paper we introduce a novel procedure for improving multiple testing
procedures (MTPs) under scenarios when the null hypothesis $p$-values tend to
be stochastically larger than standard uniform (referred to as 'inflated'). An
important class of problems for which this occurs are tests of interval
hypotheses. The new procedure starts with a set of $p$-values and discards
those with values above a certain pre-selected threshold while the rest are
corrected (scaled-up) by the value of the threshold. Subsequently, a chosen
family-wise error rate (FWER) or false discovery rate (FDR) MTP is applied to
the set of corrected $p$-values only. We prove the general validity of this
procedure under independence of $p$-values, and for the special case of the
Bonferroni method we formulate several sufficient conditions for the control of
the FWER. It is demonstrated that this 'filtering' of $p$-values can yield
considerable gains of power under scenarios with inflated null hypotheses
$p$-values.
