  In multigroup data settings with small within-group sample sizes, standard
$F$-tests of group-specific linear hypotheses can have low power, particularly
if the within-group sample sizes are not large relative to the number of
explanatory variables. To remedy this situation, in this article we derive
alternative test statistics based on information-sharing across groups. Each
group-specific test has potentially much larger power than the standard
$F$-test, while still exactly maintaining a target type I error rate if the
hypothesis for the group is true. The proposed test for a given group uses a
statistic that has optimal marginal power under a prior distribution derived
from the data of the other groups. This statistic approaches the usual
$F$-statistic as the prior distribution becomes more diffuse, but approaches a
limiting "cone" test statistic as the prior distribution becomes extremely
concentrated. We compare the power and $p$-values of the cone test to that of
the $F$-test in some high-dimensional asymptotic scenarios. An analysis of
educational outcome data is provided, demonstrating empirically that the
proposed test is more powerful than the $F$-test.
