Ewen’s sampling formula is a foundational theoretical result that connects probability and number theory with molecular genetics and molecular evolution; it was the analytical result required for testing the neutral theory of evolution, and has since been directly or indirectly utilized in a number of population genetics statistics. Ewen’s sampling formula, in turn, is deeply connected to Stirling numbers of the first kind. 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 . By reducing the calculation from a sum of terms involving Stirling numbers to a single estimate, we simultaneously improve accuracy and dramatically increase speed.

Ewen的采样公式是将概率和数论与分子遗传学和分子进化联系起来的基础理论结果。它是测试进化中立理论所需的分析结果，并且此后直接或间接地用于许多人口遗传学统计数据中。反过来，Ewen的采样公式与第一种斯特林数紧密相关。在这里，我们探讨了这些斯特林数的累积分布函数，该函数使用不完整的贝塔函数表示，可以对总和进行单个直接估计。该估计器实现了一种改进的方法，用于计算一个有用统计量Fu的渐近估计。通过将计算从涉及斯特林数的项的总和减少到单个估计，我们可以同时提高准确性并显着提高速度。