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

Hornik, Kurt ORCID: https://orcid.org/0000-0003-4198-9911 and Grün, Bettina (2012) On Maximum Likelihood Estimation of the Concentration Parameter of von Mises-Fisher Distributions. Research Report Series / Department of Statistics and Mathematics, 120. WU Vienna University of Economics and Business, Vienna.

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Abstract

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.

Item Type:	Paper
Keywords:	Maximum likelihood / Modified Bessel function ratio / Numerical approximation / von Mises-Fisher distribution
Divisions:	Departments > Finance, Accounting and Statistics > Statistics and Mathematics
Depositing User:	Josef Leydold
Date Deposited:	17 Oct 2012 07:28
Last Modified:	24 Oct 2019 13:41
FIDES Link:	https://bach.wu.ac.at/d/research/results/64639/
URI:	https://epub.wu.ac.at/id/eprint/3669

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