  Multiple testing with discrete p-values routinely arises in various
scientific endeavors. However, procedures, including the false discovery rate
(FDR) controlling Benjamini-Hochberg (BH) procedure, often used in such
settings, being developed originally for p-values with continuous
distributions, are too conservative, and so may not be as powerful as one would
hope for. Therefore, improving the BH procedure by suitably adapting it to
discrete p-values without losing its FDR control is currently an important path
of research. This paper studies the FDR control of the BH procedure when it is
applied to mid p-values and derive conditions under which it is conservative.
Our simulation study reveals that the BH procedure applied to mid p-values may
be conservative under much more general settings than characterized in this
work, and that an adaptive version of the BH procedure applied to mid p-values
is as powerful as an existing adaptive procedure based on randomized p-values.
