TY  - GEN
TI  - On Maximum Likelihood Estimation of the
Concentration Parameter of von Mises-Fisher Distributions
AB  - 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. Computational 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 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 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.
AF  - Research Report Series Institute for Statistics and Mathematics
Report 120, Vienna
UR  - http://epub.wu.ac.at/3669/
PY  - 2012-10-01
AU  - Hornik, Kurt 
AU  - Grün, Bettina 
ER -