## AbstractDuring the last decade, many approaches for improving the linear step-up procedure for FDR-control with respect to power have been worked out. Often, this has been done under asymptotic considerations with the number $n$ of hypotheses to be tested tending to infinity. At least in the class of step-up-down (SUD) tests with fixed rejection curve and under certain dependency assumptions, optimal asymptotic power can be achieved by utilizing procedures based on the asymptotically optimal rejection curve (AORC) derived in [1]. 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 multiple tests for the finite case leading to both analytical (exact critical values) and algorithmic solutions. The results rely on upper FDR-bounds for SUD-procedures proven in [1]. Main 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 [2]. Finally, we illustrate our findings with graphical representations of the resulting stepwise rejective tests and by discussing real-world application datasets. [1] Finner, H., Dickhaus, T. and Roters, M. (2008). On the False Discovery Rate and an asymptotically optimal rejection curve. The Annals of Statistics, to appear. [2] Gavrilov, Y., Benjamini, Y. and Sarkar, S. K. (2008). An adaptive step-down procedure with proven FDR control. The Annals of Statistics, to appear.
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