Multiple testing and variable selection have gained much attention in
statistical theory and methodology research. They are dealing with the same
problem of identifying the important variables among many (Jin, 2012). However,
there is little overlap in the literature. Research on variable selection has
been focusing on selection consistency, i.e., both type I and type II errors
converging to zero. This is only possible when the signals are sufficiently
strong, contrary to many modern applications. For the regime where the signals
are both rare and weak, it is inevitable that a certain amount of false
discoveries will be allowed, as long as some error rate can be controlled. In
this paper, motivated by the research by Ji and Jin (2012) and Jin (2012) in
the rare/weak regime, we extend their UPS procedure for variable selection to
multiple testing. Under certain conditions, the new UPT procedure achieves the
fastest convergence rate of marginal false non-discovery rates, while
controlling the marginal false discovery rate at any designated level α
asymptotically. Numerical results are provided to demonstrate the advantage of
the proposed method.Comment: 27 page