Watterson estimator (${\hat{\theta }}_W=\frac{S}{{\sum }_{i=1}^{n-1}1/i}$)

For a neutral sample of size $n\ge 2$ under the infinite sites mutation model, the Watterson estimator is ${\hat{\theta }}_W=S/a_{n-1}$, where $a_{n-1}={\sum }_{i=1}^{n-1}1/i$. Since the coalescent has $ES=\theta a_{n-1}$, this estimator is unbiased. It matches the observed segregating-site count to its expected value, but need not maximize the exact segregating-site likelihood under the neutral coalescent.

For fixed $\theta >0$, a geometric mutation count in a coalescent epoch with mean ${\mu }_i=\theta /i$ has third moment ${\mu }_i+6{\mu }_i^2+6{\mu }_i^3$. The centered absolute third moment is at most $4\mathbb{E}[Y_i^3]+4{\mu }_i^3$, so its sum is $O(\log n)$. The total variance is asymptotic to $\theta \log n$, making the Lyapunov condition with $\delta =1$ hold. Rescaling by the harmonic sum gives the displayed limit. At $\theta =0$ the estimator is identically zero and the limit is degenerate. The variance of the Watterson estimator decreases only logarithmically with sample size at a fixed nonrecombining locus.

The moments of the mutation count on a neutral coalescent tree give this variance for ${\hat{\theta }}_W=S/H_{n-1}$. Since $H_{n-1}\sim \log n$ and the sum of inverse squares is bounded, it tends to zero. Together with unbiasedness, this gives mean-square convergence to the mutation parameter, so it is a consistent estimator.