  Improved procedures, in terms of smaller missed discovery rates (MDR), for
performing multiple hypotheses testing with weak and strong control of the
family-wise error rate (FWER) or the false discovery rate (FDR) are developed
and studied. The improvement over existing procedures such as the \v{S}id\'ak
procedure for FWER control and the Benjamini--Hochberg (BH) procedure for FDR
control is achieved by exploiting possible differences in the powers of the
individual tests. Results signal the need to take into account the powers of
the individual tests and to have multiple hypotheses decision functions which
are not limited to simply using the individual $p$-values, as is the case, for
example, with the \v{S}id\'ak, Bonferroni, or BH procedures. They also enhance
understanding of the role of the powers of individual tests, or more precisely
the receiver operating characteristic (ROC) functions of decision processes, in
the search for better multiple hypotheses testing procedures. A
decision-theoretic framework is utilized, and through auxiliary randomizers the
procedures could be used with discrete or mixed-type data or with rank-based
nonparametric tests. This is in contrast to existing $p$-value based procedures
whose theoretical validity is contingent on each of these $p$-value statistics
being stochastically equal to or greater than a standard uniform variable under
the null hypothesis. Proposed procedures are relevant in the analysis of
high-dimensional "large $M$, small $n$" data sets arising in the natural,
physical, medical, economic and social sciences, whose generation and creation
is accelerated by advances in high-throughput technology, notably, but not
limited to, microarray technology.
