  Hypothesis testing is a key part of empirical science and multiple testing as
well as the combination of evidence from several tests are continued areas of
research. In this article we consider the problem of combining the results of
multiple hypothesis tests to i) test global hypotheses and ii) make marginal
inference while controlling the k-FWER. We propose a new family of combination
tests for joint hypotheses, called the 'Too Many, Too Improbable' (TMTI)
statistics, which we show through simulation to have higher power than other
combination tests against many alternatives. Furthermore, we prove that a large
family of combination tests -- which includes the one we propose but also other
combination tests -- admits a quadratic shortcut when used in a Closed Testing
Procedure, which controls the FWER strongly. We develop an algorithm that is
linear in the number of hypotheses for obtaining confidence sets for the number
of false hypotheses among a collection of hypotheses and an algorithm that is
cubic in the number of hypotheses for controlling the k-FWER for any k greater
than one.
