 Solution

The asymptotic normality of the Watterson estimator gives approximate standard error $\sqrt{\theta /\log n}$. Thus precision at a fixed nonrecombining locus improves only logarithmically with sample size. To halve the asymptotic standard error, one needs $\log n^{\prime }\simeq 4\log n$, or roughly $n^{\prime }=n^4$, rather than four times as many sampled chromosomes.

For large samples and positive $\hat{\theta }$, a plug-in normal approximation gives $\hat{\theta }\pm 1.96\sqrt{\hat{\theta }/\log n}$ as an approximate 95% confidence interval. At low mutation counts it can perform poorly or give a negative endpoint; the exact count model is then preferable. Multiple independent, comparable loci can provide additional independent information, whereas tightly linked chromosomes still share the random ancestry at one locus. Genetic recombination, nonneutral evolution and changing population size require corresponding changes to the model.