  Multivariate statistics are often available as well as necessary in
hypothesis tests. We study how to use such statistics to control not only false
discovery rate (FDR) but also positive FDR (pFDR) with good power. We show that
FDR can be controlled through nested regions of multivariate $p$-values of test
statistics. If the distributions of the test statistics are known, then the
regions can be constructed explicitly to achieve FDR control with maximum power
among procedures satisfying certain conditions. On the other hand, our focus is
where the distributions are only partially known. Under certain conditions, a
type of nested regions are proposed and shown to attain (p)FDR control with
asymptotically maximum power as the pFDR control level approaches its
attainable limit. The procedure based on the nested regions is compared with
those based on other nested regions that are easier to construct as well as
those based on more straightforward combinations of the test statistics.
