Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio \(R_\nu = I_{\nu +1} / I_\nu \) of modified Bessel functions and computational methods are required to invert these functions using approximative or iterative algorithms. In this paper we use Amos-type bounds for \(R_\nu \) to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of \(R_\nu \) is evaluated at values tending to \(1\) (from the left). We show that previously introduced rational bounds for \(R_\nu \) which are invertible using quadratic equations cannot be used to improve these bounds.

中文翻译：

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关于 von Mises-Fisher 分布浓度参数的最大似然估计。

von Mises-Fisher 分布的浓度参数的最大似然估计涉及对修正贝塞尔函数的比值\(R_\nu = I_{\nu +1} / I_\nu \)求反，需要使用计算方法来反演这些函数近似或迭代算法。在本文中，我们使用\(R_\nu \) 的Amos 类型边界来推导出更清晰的反函数边界，确定这些边界的近似误差，并使用这些来提出一个新的近似值，当其误差趋于零时\(R_\nu \)的倒数在趋向于\(1\) 的值处进行评估（从左侧开始）。我们证明了之前为\(R_\nu \)引入的有理边界 使用二次方程可逆的不能用于改进这些界限。