  In this work we study an adaptive step-down procedure for testing $m$
hypotheses. It stems from the repeated use of the false discovery rate
controlling the linear step-up procedure (sometimes called BH), and makes use
of the critical constants $iq/[(m+1-i(1-q)]$, $i=1,...,m$. Motivated by its
success as a model selection procedure, as well as by its asymptotic
optimality, we are interested in its false discovery rate (FDR) controlling
properties for a finite number of hypotheses. We prove this step-down procedure
controls the FDR at level $q$ for independent test statistics. We then
numerically compare it with two other procedures with proven FDR control under
independence, both in terms of power under independence and FDR control under
positive dependence.
