TY - UNPB
T1 - On Maximum Likelihood Estimation of the
Concentration Parameter of von Mises-Fisher Distributions
AU - Hornik, Kurt
AU - Grün, Bettina
PY - 2012/10/1
Y1 - 2012/10/1
N2 - 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.
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.
U2 - 10.57938/744b188f-4524-4ede-8ab1-210749c2b8ee
DO - 10.57938/744b188f-4524-4ede-8ab1-210749c2b8ee
M3 - WU Working Paper
T3 - Research Report Series / Department of Statistics and Mathematics
BT - On Maximum Likelihood Estimation of the
Concentration Parameter of von Mises-Fisher Distributions
ER -