On Maximum Likelihood Estimation of the Concentration Parameter of von Mises-Fisher

Maximum likelihood estimation of the concentration parameter of von Mises-Fisher distributions involves inverting the ratio Rν = Iν+1/Iν of modified Bessel functions. Com-putational issues when using approximative or iterative methods were discussed in Tanabe et al. (Comput Stat 22(1):145–157, 2007) and Sra (Comput Stat 27(1):177–190, 2012). 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