  A novel class of methods for combining $p$-values to perform aggregate
hypothesis tests has emerged that exploit the properties of heavy-tailed Stable
distributions. These methods offer important practical advantages including
robustness to dependence and better-than-Bonferroni scaleability, and they
reveal theoretical connections between Bayesian and classical hypothesis tests.
The harmonic mean $p$-value (HMP) procedure is based on the convergence of
summed inverse $p$-values to the Landau distribution, while the Cauchy
combination test (CCT) is based on the self-similarity of summed
Cauchy-transformed $p$-values. The CCT has the advantage that it is analytic
and exact. The HMP has the advantage that it emulates a model-averaged Bayes
factor, is insensitive to $p$-values near 1, and offers multilevel testing via
a closed testing procedure. Here I investigate whether other Stable combination
tests can combine these benefits, and identify a new method, the L\'evy
combination test (LCT). The LCT exploits the self-similarity of sums of L\'evy
random variables transformed from $p$-values. Under arbitrary dependence, the
LCT possesses better robustness than the CCT and HMP, with two-fold worst-case
inflation at small significance thresholds. It controls the strong-sense
familywise error rate through a multilevel test uniformly more powerful than
Bonferroni. Simulations show that the LCT behaves like Simes' test in some
respects, with power intermediate between the HMP and Bonferroni. The LCT
represents an interesting and attractive addition to combined testing methods
based on heavy-tailed distributions.
