IDEAS home Printed from
   My bibliography  Save this paper

Inference for Spherical Location under High Concentration


  • Davy Paindaveine
  • Thomas Verdebout


Motivated by the fact that circular or spherical data are often much concentrated around a location θ, we consider inference about θunder high concentration asymptotic scenarios for which the probability of any fixed spherical cap centered at θ converges to one as the sample size n diverges to infinity. Rather than restricting to Fisher– von Mises–Langevin distributions, we consider a much broader, semiparametric, class of rotationally symmetric distributions indexed by the location parameter θ, a scalar concentration parameter κ and a functional nuisance f. We determine the class of distributions for which high concentration is obtained as κ diverges to infinity. For such distributions, we then consider inference (point estimation, confidence zone estimation, hypothesis testing) on θ in asymptotic scenarios where κn diverges to infinity at an arbitrary rate with the sample size n. Our asymptotic investigation reveals that, interestingly, optimal inference procedures on θ show consistency rates that depend on f. Using asymptotics “`a la Le Cam”, we show that the spherical mean is, at any f, a parametrically super-efficient estimator of θ and that the Watson and Wald tests for H0 :θ = θ0 enjoy similar, non-standard, optimality properties. Our results are illustrated by Monte Carlo simulations. On a technical point of view, our asymptotic derivations require challenging expansions of rotationally symmetric functionals for large arguments of the nuisance function f.

Suggested Citation

  • Davy Paindaveine & Thomas Verdebout, 2019. "Inference for Spherical Location under High Concentration," Working Papers ECARES 2019-02, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/280743

    Download full text from publisher

    File URL:
    File Function: Œuvre complète ou partie de l'œuvre
    Download Restriction: no

    References listed on IDEAS

    1. Michael Rosenthal & Wei Wu & Eric Klassen & Anuj Srivastava, 2014. "Spherical Regression Models Using Projective Linear Transformations," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(508), pages 1615-1624, December.
    2. SenGupta, Ashis & Kim, Sungsu & Arnold, Barry C., 2013. "Inverse circular–circular regression," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 200-208.
    3. T. Hayakawa, 1990. "On tests for the mean direction of the Langevin distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 359-373, June.
    4. Chikuse, Yasuko, 2003. "Concentrated matrix Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 85(2), pages 375-394, May.
    5. Louis-Paul Rivest & Thierry Duchesne & Aurélien Nicosia & Daniel Fortin, 2016. "A general angular regression model for the analysis of data on animal movement in ecology," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 65(3), pages 445-463, April.
    6. Fujikoshi, Yasunori & Watamori, Yoko, 1992. "Tests for the mean direction of the Langevin distribution with large concentration parameter," Journal of Multivariate Analysis, Elsevier, vol. 42(2), pages 210-225, August.
    7. Watson, Geoffrey S., 1984. "The theory of concentrated Langevin distributions," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 74-82, February.
    8. T. D. Downs, 2003. "Spherical regression," Biometrika, Biometrika Trust, vol. 90(3), pages 655-668, September.
    9. Jupp, P.E., 2015. "Copulae on products of compact Riemannian manifolds," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 92-98.
    10. R. Arnold & P. E. Jupp, 2013. "Statistics of orthogonal axial frames," Biometrika, Biometrika Trust, vol. 100(3), pages 571-586.
    11. Arnold, R. & Jupp, P.E. & Schaeben, H., 2018. "Statistics of ambiguous rotations," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 73-85.
    Full references (including those not matched with items on IDEAS)


    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:eca:wpaper:2013/280743. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Benoit Pauwels). General contact details of provider: .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.