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Stirling numbers of the first kind are common in number theory andcombinatorics; through Ewen's sampling formula, these numbers enter into thecalculation of several population genetics statistics, such as Fu's Fs. Inprevious papers we have considered an asymptotic estimator for a finite sum ofStirling numbers, which enables rapid and accurate calculation of Fu's Fs.These sums can also be viewed as a cumulative distribution function; thisformulation leads directly to an inversion problem, where, given a value forFu's Fs, the goal is to solve for one of the input parameters. We solve thisinversion using Newton iteration for small parameters. For large parameters weneed to extend the earlier obtained asymptotic results to handle the inversionproblem asymptotically. Numerical experiments are given to show the efficiencyof both solving the inversion problem and the expanded estimator for thestatistical quantities.

A distribution function from population genetics statistics using  Stirling numbers of the first kind: Asymptotics, inversion and numerical  evaluation