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

TL;DR

This paper develops asymptotic methods and numerical techniques to efficiently evaluate and invert a distribution function derived from Stirling numbers of the first kind, which are used in population genetics statistics like Fu's Fs.

Contribution

It introduces an asymptotic estimator for sums of Stirling numbers, extends it for inversion problems, and demonstrates efficient numerical solutions for population genetics applications.

Findings

Asymptotic estimator enables rapid calculation of Fu's Fs.

Newton iteration effectively solves the inversion problem for small parameters.

Extended asymptotic results improve inversion accuracy for large parameters.

Abstract

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.