  Simultaneous inference allows for the exploration of data while deciding on
criteria for proclaiming discoveries. It was recently proved that all
admissible post-hoc inference methods for true discoveries must employ closed
testing. In this paper, we investigate efficient closed testing with local
tests of a special form: thresholding a function of sums of test scores for the
individual hypotheses. Under this special design, we propose a new statistic
that quantifies the cost of multiplicity adjustments, and we develop fast
(mostly linear-time) algorithms for post-hoc inference. Paired with recent
advances in global null tests based on generalized means, our work instantiates
a series of simultaneous inference methods that can handle many dependence
structures and signal compositions. We provide guidance on the method choices
via theoretical investigation of the conservativeness and sensitivity for
different local tests, as well as simulations that find analogous behavior for
local tests and full closed testing.
