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On maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions

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  • Kurt Hornik
  • Bettina Grün

Abstract

Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio $$R_\nu=I_{\nu +1} / I_\nu $$ R ν = I ν + 1 / I ν of modified Bessel functions and computational methods are required to invert these functions using approximative or iterative algorithms. In this paper we use Amos-type bounds for $$R_\nu $$ R ν 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 $$R_\nu $$ R ν is evaluated at values tending to $$1$$ 1 (from the left). We show that previously introduced rational bounds for $$R_\nu $$ R ν which are invertible using quadratic equations cannot be used to improve these bounds. Copyright The Author(s) 2014

Suggested Citation

  • Kurt Hornik & Bettina Grün, 2014. "On maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions," Computational Statistics, Springer, vol. 29(5), pages 945-957, October.
    • Handle: RePEc:spr:compst:v:29:y:2014:i:5:p:945-957
    DOI: 10.1007/s00180-013-0471-0
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    References listed on IDEAS

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    1. Suvrit Sra, 2012. "A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of I s (x)," Computational Statistics, Springer, vol. 27(1), pages 177-190, March.
    2. Lin Yuan & John Kalbfleisch, 2000. "On the Bessel Distribution and Related Problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 438-447, September.
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    Cited by:

    1. Xavier Bry & Lionel Cucala, 2022. "A von Mises–Fisher mixture model for clustering numerical and categorical variables," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(2), pages 429-455, June.

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