  In the context of multiple hypotheses testing, the proportion $\pi_0$ of true
null hypotheses in the pool of hypotheses to test often plays a crucial role,
although it is generally unknown a priori. A testing procedure using an
implicit or explicit estimate of this quantity in order to improve its
efficency is called adaptive. In this paper, we focus on the issue of False
Discovery Rate (FDR) control and we present new adaptive multiple testing
procedures with control of the FDR. First, in the context of assuming
independent $p$-values, we present two new procedures and give a unified review
of other existing adaptive procedures that have provably controlled FDR. We
report extensive simulation results comparing these procedures and testing
their robustness when the independence assumption is violated. The new proposed
procedures appear competitive with existing ones. The overall best, though, is
reported to be Storey's estimator, but for a parameter setting that does not
appear to have been considered before. Second, we propose adaptive versions of
step-up procedures that have provably controlled FDR under positive dependences
and unspecified dependences of the $p$-values, respectively. While simulations
only show an improvement over non-adaptive procedures in limited situations,
these are to our knowledge among the first theoretically founded adaptive
multiple testing procedures that control the FDR when the $p$-values are not
independent.
