On maximum likelihood estimation of the concentration parameter of von Mises-Fisher distributions.

Maximum likelihood estimation of the concentration parameter of von
Mises–Fisher distributions involves inverting the ratio Rν = Iν+1/Iν 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 Rν 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 Rν is evaluated at values tending to 1 (from the
left). We show that previously introduced rational bounds for Rν which are invertible
using quadratic equations cannot be used to improve these bounds.