Quotation Hornik, Kurt, Grün, Bettina. 2012. On Maximum Likelihood Estimation of the Concentration Parameter of von Mises-Fisher Distributions. Research Report Series Institute for Statistics and Mathematics Report 120, 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.

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Publication's profile

Status of publication Published
Affiliation WU
Type of publication Working/discussion paper, preprint
Language English
Title On Maximum Likelihood Estimation of the Concentration Parameter of von Mises-Fisher Distributions
Title of whole publication Research Report Series Institute for Statistics and Mathematics Report 120, Vienna
Year 2012
URL http://epub.wu.ac.at/3669/

Associations

People
Hornik, Kurt (Details)
Grün, Bettina (Details)
Organization
Institute for Statistics and Mathematics IN (Details)
Research Institute for Computational Methods FI (Details)
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