False discovery rate envelopes

False discovery rate (FDR) is a common way to control the number of false discoveries in multiple testing. In this paper, the focus is on functional test statistics, which are discretized into $m$ highly correlated hypotheses. The aim is to find, based on resampling principles, a graphical envelope that detects the outcomes of all individual hypotheses by a simple rule: the hypothesis is rejected if and only if the empirical test statistic is outside of the envelope. Such an envelope offers a straightforward interpretation of the test results similarly as in global envelope testing recently developed with controlling the family-wise error rate. Two different adaptive single threshold procedures are developed to fulfill this aim. The new methods are illustrated by three real data examples.