Strict FDR-control based on the asymptotically optimal rejection curve
Recently, many approaches for improving the linear step-up procedure with respect to power have been worked out. Often, this has been done under asymptotic considerations with the number $n$ of hypotheses tending to infinity. In the class of step-up-down (SUD) tests with fixed rejection curve and under certain dependency assumptions, optimal asymptotic power can be achieved with procedures based on the asymptotically optimal rejection curve (AORC) derived in . However, for fixed $n$, procedures based on the AORC often violate the pre-specified FDR-level. Therefore, we present and discuss modifications of AORC-based tests for the finite case leading to both analytical and algorithmic solutions. The results rely on upper FDR-bounds proven in . Theoretical tools for deriving these bounds are structural properties of SUD tests. For evaluation of the resulting formulas, efficient recursive algorithms for computing the joint distribution function of order statistics will be employed. It is remarkable that a step-down procedure with AORC-related critical values has proven FDR-control for any finite $n$ as recently shown in . Finally, we illustrate our findings with graphical representations of the resulting stepwise rejective tests and by discussing real-world application datasets.