  Stirling numbers of the first kind are used in the derivation of several
population genetics statistics, which in turn are useful for testing
evolutionary hypotheses directly from DNA sequences. Here, we explore the
cumulative distribution function of these Stirling numbers, which enables a
single direct estimate of the sum, using representations in terms of the
incomplete beta function. This estimator enables an improved method for
calculating an asymptotic estimate for one useful statistic, Fu's $F_s$. By
reducing the calculation from a sum of terms involving Stirling numbers to a
single estimate, we simultaneously improve accuracy and dramatically increase
speed.
