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

Too Many, Too Improbable: testing joint hypotheses and closed testing shortcuts