Truncated Simes tests

It is known that the one-sided Simes' test controls the error rate if the underlying distribution is multivariate totally positive of order 2 (MTP2), but not in general. The two-sided test also controls the error rate when the coordinate absolute value has an MTP2 distribution, which holds more generally. We prove mathematically that when the coordinate absolute value controls the error rate at level 2α, then certain kinds of truncated Simes' tests also control the one-sided error rate at level α. We also compare the closure of the truncated tests with the Holms, Hochberg, and Hommel procedures in many scenarios when the test statistics are multivariate normal.