  The severity of type II errors is frequently ignored when deriving a multiple
testing procedure, even though utilizing it properly can greatly help in making
correct decisions. This paper puts forward a theory behind developing a
multiple testing procedure that can incorporate the type II error severity and
is optimal in the sense of minimizing a measure of false non-discoveries among
all procedures controlling a measure of false discoveries. The theory is
developed under a general model allowing arbitrary dependence by taking a
compound decision theoretic approach to multiple testing with a loss function
incorporating the type II error severity. We present this optimal procedure in
its oracle form and offer numerical evidence of its superior performance over
relevant competitors.
