  Much effort has been done to control the "false discovery rate" (FDR) when
$m$ hypotheses are tested simultaneously. The FDR is the expectation of the
"false discovery proportion" $\text{FDP}=V/R$ given by the ratio of the number
of false rejections $V$ and all rejections $R$. In this paper, we have a closer
look at the FDP for adaptive linear step-up multiple tests. These tests extend
the well known Benjamini and Hochberg test by estimating the unknown amount
$m_0$ of the true null hypotheses. We give exact finite sample formulas for
higher moments of the FDP and, in particular, for its variance. Using these
allows us a precise discussion about the consistency of adaptive step-up tests.
We present sufficient and necessary conditions for consistency on the
estimators $\widehat m_0$ and the underlying probability regime. We apply our
results to convex combinations of generalized Storey type estimators with
various tuning parameters and (possibly) data-driven weights. The corresponding
step-up tests allow a flexible adaptation. Moreover, these tests control the
FDR at finite sample size. We compare these tests to the classical Benjamini
and Hochberg test and discuss the advantages of it.
