In this paper we introduce and investigate a new rejection curve for asymptotic control of the false discovery rate (FDR) in multiple hypotheses testing problems. We first give a heuristic motivation for this new curve and propose some procedures related to it. Then we introduce a set of possible assumptions and give a unifying short proof of FDR control for procedures based on Simes' critical values, whereby certain types of dependency are allowed. This methodology of proof is then applied to other fixed rejection curves including the proposed new curve. Among others, we investigate the problem of finding least favorable parameter configurations such that the FDR becomes largest. We then derive a series of results concerning asymptotic FDR control for procedures based on the new curve and discuss several example procedures in more detail. A main result will be an asymptotic optimality statement for various procedures based on the new curve in the class of fixed rejection curves. Finally, we briefly discuss strict FDR control for a finite number of hypotheses.
Some effort has been undertaken over the last decade to provide conditions
for the control of the false discovery rate by the linear step-up procedure
(LSU) for testing $n$ hypotheses when test statistics are dependent. In this
paper we investigate the expected error rate (EER) and the false discovery rate
(FDR) in some extreme parameter configurations when $n$ tends to infinity for
test statistics being exchangeable under null hypotheses. All results are
derived in terms of $p$-values. In a general setup we present a series of
results concerning the interrelation of Simes' rejection curve and the
(limiting) empirical distribution function of the $p$-values. Main objects
under investigation are largest (limiting) crossing points between these
functions, which play a key role in deriving explicit formulas for EER and FDR.
As specific examples we investigate equi-correlated normal and $t$-variables in
more detail and compute the limiting EER and FDR theoretically and numerically.
A surprising limit behavior occurs if these models tend to independence.Comment: Published in at http://dx.doi.org/10.1214/009053607000000046 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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