  The False Discovery Rate (FDR) paradigm aims to attain certain control on
Type I errors with relatively high power for multiple hypothesis testing. The
Benjamini--Hochberg (BH) procedure is a well-known FDR controlling procedure.
Under a random effects model, we show that, in general, unlike the FDR, the
positive FDR (pFDR) of the BH procedure cannot be controlled at an arbitrarily
low level due to the limited evidence provided by the observations to separate
false and true nulls. This results in a criticality phenomenon, which is
characterized by a transition of the procedure's power from being positive to
asymptotically 0 without any reduction in the pFDR, once the target FDR control
level is below a positive critical value. To address the constraints on the
power and pFDR control imposed by the criticality phenomenon, we propose a
procedure which applies BH-type procedures at multiple locations in the domain
of $p$-values. Both analysis and simulations show that the proposed procedure
can attain substantially improved power and pFDR control.
