  We theoretically analyze the problem of testing for $p$-hacking based on
distributions of $p$-values across multiple studies. We provide general results
for when such distributions have testable restrictions (are non-increasing)
under the null of no $p$-hacking. We find novel additional testable
restrictions for $p$-values based on $t$-tests. Specifically, the shape of the
power functions results in both complete monotonicity as well as bounds on the
distribution of $p$-values. These testable restrictions result in more powerful
tests for the null hypothesis of no $p$-hacking. When there is also publication
bias, our tests are joint tests for $p$-hacking and publication bias. A
reanalysis of two prominent datasets shows the usefulness of our new tests.
