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