  The most popular multiple testing procedures are stepwise procedures based on
$P$-values for individual test statistics. Included among these are the false
discovery rate (FDR) controlling procedures of Benjamini--Hochberg [J. Roy.
Statist. Soc. Ser. B 57 (1995) 289--300] and their offsprings. Even for models
that entail dependent data, $P$-values based on marginal distributions are
used. Unlike such methods, the new method takes dependency into account at all
stages. Furthermore, the $P$-value procedures often lack an intuitive convexity
property, which is needed for admissibility. Still further, the new methodology
is computationally feasible. If the number of tests is large and the proportion
of true alternatives is less than say 25 percent, simulations demonstrate a
clear preference for the new methodology. Applications are detailed for models
such as testing treatments against control (or any intraclass correlation
model), testing for change points and testing means when correlation is
successive.
