  We introduce a low dimensional function of the site frequency spectrum that
is tailor-made for distinguishing coalescent models with multiple mergers from
Kingman coalescent models with population growth, and use this function to
construct a hypothesis test between these model classes. The null and
alternative sampling distributions of the statistic are intractable, but its
low dimensionality renders them amenable to Monte Carlo estimation. We
construct kernel density estimates of the sampling distributions based on
simulated data, and show that the resulting hypothesis test dramatically
improves on the statistical power of a current state-of-the-art method. A key
reason for this improvement is the use of multi-locus data, in particular
averaging observed site frequency spectra across unlinked loci to reduce
sampling variance. We also demonstrate the robustness of our method to nuisance
and tuning parameters. Finally we show that the same kernel density estimates
can be used to conduct parameter estimation, and argue that our method is
readily generalisable for applications in model selection, parameter inference
and experimental design.
