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Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants

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  • A. Kume
  • Andrew T. A. Wood

Abstract

The Fisher--Bingham distribution is obtained when a multivariate normal random vector is conditioned to have unit length. Its normalising constant can be expressed as an elementary function multiplied by the density, evaluated at 1, of a linear combination of independent noncentral χ-sub-1-super-2 random variables. Hence we may approximate the normalising constant by applying a saddlepoint approximation to this density. Three such approximations, implementation of each of which is straightforward, are investigated: the first-order saddlepoint density approximation, the second-order saddlepoint density approximation and a variant of the second-order approximation which has proved slightly more accurate than the other two. The numerical and theoretical results we present showthat this approach provides highly accurate approximations in a broad spectrum of cases. Copyright 2005, Oxford University Press.

Suggested Citation

  • A. Kume & Andrew T. A. Wood, 2005. "Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants," Biometrika, Biometrika Trust, vol. 92(2), pages 465-476, June.
    • Handle: RePEc:oup:biomet:v:92:y:2005:i:2:p:465-476
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    File URL: http://hdl.handle.net/10.1093/biomet/92.2.465
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    Cited by:

    1. Ley, Christophe & Verdebout, Thomas, 2017. "Skew-rotationally-symmetric distributions and related efficient inferential procedures," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 67-81.
    2. Tamio Koyama & Akimichi Takemura, 2016. "Holonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables," Computational Statistics, Springer, vol. 31(4), pages 1645-1659, December.
    3. Kume, A. & Wood, Andrew T.A., 2007. "On the derivatives of the normalising constant of the Bingham distribution," Statistics & Probability Letters, Elsevier, vol. 77(8), pages 832-837, April.
    4. Christophe Ley & Thomas Verdebout, 2014. "Skew-rotsymmetric Distributions on Unit Spheres and Related Efficient Inferential Proceedures," Working Papers ECARES ECARES 2014-46, ULB -- Universite Libre de Bruxelles.
    5. Jean-Luc Dortet-Bernadet & Nicolas Wicker, 2018. "A Note on Inverse Stereographic Projection of Elliptical Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 138-151, February.
    6. Bee, Marco & Benedetti, Roberto & Espa, Giuseppe, 2017. "Approximate maximum likelihood estimation of the Bingham distribution," Computational Statistics & Data Analysis, Elsevier, vol. 108(C), pages 84-96.
    7. Takasu, Yuya & Yano, Keisuke & Komaki, Fumiyasu, 2018. "Scoring rules for statistical models on spheres," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 111-115.
    8. Ian L. Dryden, 2014. "Comment on Geodesic Monte Carlo on Embedded Manifolds by Byrne and Girolami," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 8-9, March.
    9. Marco Bee & Roberto Benedetti & Giuseppe Espa, 2015. "Approximate likelihood inference for the Bingham distribution," DEM Working Papers 2015/02, Department of Economics and Management.
    10. Sei, Tomonari & Shibata, Hiroki & Takemura, Akimichi & Ohara, Katsuyoshi & Takayama, Nobuki, 2013. "Properties and applications of Fisher distribution on the rotation group," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 440-455.
    11. Kume, A. & Walker, S.G., 2014. "On the Bingham distribution with large dimension," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 345-352.
    12. Tamio Koyama & Hiromasa Nakayama & Kenta Nishiyama & Nobuki Takayama, 2014. "Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$ d -dimensional sphere," Computational Statistics, Springer, vol. 29(3), pages 661-683, June.
    13. Tianlu Yuan, 2021. "The 8-parameter Fisher–Bingham distribution on the sphere," Computational Statistics, Springer, vol. 36(1), pages 409-420, March.

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