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

Publication: Working/Discussion PaperWU Working Paper

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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.
Original languageEnglish
Publication statusPublished - 1 Oct 2012

Publication series

SeriesResearch Report Series / Department of Statistics and Mathematics

WU Working Paper Series

  • Research Report Series / Department of Statistics and Mathematics

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