  In the multiple testing problem with independent tests, the classical linear
step-up procedure controls the false discovery rate (FDR) at level
$\pi_0\alpha$, where $\pi_0$ is the proportion of true null hypotheses and
$\alpha$ is the target FDR level. Adaptive procedures can improve power by
incorporating estimates of $\pi_0$, which typically rely on a tuning parameter.
Fixed adaptive procedures set their tuning parameters before seeing the data
and can be shown to control the FDR in finite samples. We develop theoretical
results for dynamic adaptive procedures whose tuning parameters are determined
by the data. We show that, if the tuning parameter is chosen according to a
left-to-right stopping time rule, the corresponding dynamic adaptive procedure
controls the FDR in finite samples. Examples include the recently proposed
right-boundary procedure and the widely used lowest-slope procedure, among
others. Simulation results show that the right-boundary procedure is more
powerful than other dynamic adaptive procedures under independence and mild
dependence conditions.
