  This paper considers the problem of testing many moment inequalities, where
the number of moment inequalities ($p$) is possibly larger than the sample size
($n$). Chernozhukov et al. (2019) proposed asymptotic tests for this problem
using the maximum $t$ statistic. We observe that such tests can have low power
if multiple inequalities are violated. As an alternative, we propose novel
randomization tests based on a maximum non-negatively weighted combination of
$t$ statistics. We provide a condition guaranteeing size control in large
samples. Simulations show that the tests control size in small samples ($n =
30$, $p = 1000$), and often has substantially higher power against alternatives
with multiple violations than tests based on the maximum $t$ statistic.
