  In multiple hypothesis testing, the volume of data, defined as the number of
replications per null times the total number of nulls, usually defines the
amount of resource required. On the other hand, power is an important measure
of performance for multiple testing. Due to practical constraints, the number
of replications per null may not be large enough in terms of the difference
between false and true nulls. For the case where the population fraction of
false nulls is constant, we show that, as the difference between false and true
nulls becomes increasingly subtle while the number of replications per null
cannot increase fast enough, (1) in order to have enough chance that the data
to be collected will yield some trustworthy findings, as measured by a
conditional version of the positive false discovery rate (pFDR), the volume of
data has to grow at a rate much faster than in the case where the number of
replications per null can be large enough, and (2) in order to control the pFDR
asymptotically, power has to decay to 0 in a rate highly sensitive to rejection
criterion and there is no asymptotically most powerful procedures among those
that control the pFDR asymptotically at the same level.
