IDEAS home Printed from https://ideas.repec.org/a/spr/compst/v29y2014i5p945-957.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00180-013-0471-0
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00180-013-0471-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. T. Pellegrino & P. Sabino, 2015. "Enhancing Least Squares Monte Carlo with diffusion bridges: an application to energy facilities," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 761-772, May.
    2. Roberto León-González, 2019. "Efficient Bayesian inference in generalized inverse gamma processes for stochastic volatility," Econometric Reviews, Taylor & Francis Journals, vol. 38(8), pages 899-920, September.
    3. Hornik, Kurt & Grün, Bettina, 2014. "movMF: An R Package for Fitting Mixtures of von Mises-Fisher Distributions," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 58(i10).
    4. You, Kisung & Suh, Changhee, 2022. "Parameter estimation and model-based clustering with spherical normal distribution on the unit hypersphere," Computational Statistics & Data Analysis, Elsevier, vol. 171(C).
    5. Makarov Roman N. & Glew Devin, 2010. "Exact simulation of Bessel diffusions," Monte Carlo Methods and Applications, De Gruyter, vol. 16(3-4), pages 283-306, January.
    6. Minerva Mukhopadhyay & Didong Li & David B. Dunson, 2020. "Estimating densities with non‐linear support by using Fisher–Gaussian kernels," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1249-1271, December.
    7. Biswas, Atanu & Jha, Jayant & Dutta, Somak, 2016. "Modelling circular random variables with a spike at zero," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 194-201.
    8. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
    9. Árpád Baricz, 2014. "Remarks on a parameter estimation for von Mises–Fisher distributions," Computational Statistics, Springer, vol. 29(3), pages 891-894, June.
    10. Xu, Hang & Alvo, Mayer & Yu, Philip L.H., 2018. "Angle-based models for ranking data," Computational Statistics & Data Analysis, Elsevier, vol. 121(C), pages 113-136.
    11. Devroye, Luc, 2002. "Simulating Bessel random variables," Statistics & Probability Letters, Elsevier, vol. 57(3), pages 249-257, April.
    12. Jan Baldeaux & Dale Roberts, 2012. "Quasi-Monte Carol Methods for the Heston Model," Research Paper Series 307, Quantitative Finance Research Centre, University of Technology, Sydney.
    13. Ferdinand Bollwein & Stephan Westphal, 2022. "Oblique decision tree induction by cross-entropy optimization based on the von Mises–Fisher distribution," Computational Statistics, Springer, vol. 37(5), pages 2203-2229, November.
    14. Ian Iscoe & Asif Lakhany, 2011. "Adaptive Simulation of the Heston Model," Papers 1111.6067, arXiv.org.
    15. S. T. Tse & Justin W. L. Wan, 2013. "Low-bias simulation scheme for the Heston model by Inverse Gaussian approximation," Quantitative Finance, Taylor & Francis Journals, vol. 13(6), pages 919-937, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:compst:v:29:y:2014:i:5:p:945-957. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.