title: On maximum likelihood estimation of the concentration parameter of von Mises-Fisher distributions.
abstract: Maximum likelihood estimation of the concentration parameter of von Mises-Fisher distributions involves inverting the ratio [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] is evaluated at values tending to [Formula: see text] (from the left). We show that previously introduced rational bounds for [Formula: see text] which are invertible using quadratic equations cannot be used to improve these bounds.
authors.affiliation: Institute for Statistics and Mathematics, WU Wirtschaftsuniversität Wien, Vienna, Austria.
authors.forename: Kurt
authors.initials: K
authors.lastname: Hornik
authors.affiliation: Institut für Angewandte Statistik, Johannes Kepler Universität Linz, Linz, Austria.
authors.forename: Bettina
authors.initials: B
authors.lastname: Grün
keywords: Maximum likelihood; Modified Bessel function ratio ; Numerical approximation; von Mises–Fisher distribution