Sufficiency of the number of segregating sites in the limit under finite-sites mutation

We show that the number of segregating sites is a sufficient statistic for the scaled mutation parameter (θ) in the limit as the number of sites tends to infinity and there is free recombination between sites. We assume that the mutation parameter at each site tends to zero such than the total mutation parameter (θ) is constant in the limit. Our results show that Watterson’s estimator is the maximum likelihood estimator in this case, but that it estimates a composite parameter which is different for different mutation models. Some of our results hold when recombination is limited, because Watterson’s estimator is an unbiased, method-of-moments estimator regardless of the recombination rate. The quantity it estimates depends on the details of how mutations occur at each site.