IDEAS home Printed from
   My bibliography  Save this paper

Detecting the Direction of a Signal on High-dimensional Spheres: Non-null and Le Cam Optimality Results


  • Davy Paindaveine
  • Thomas Verdebout


We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction theta of a Fisher-von Mises-Langevin distribution on the p-dimensional unit hypersphere is equal to a given direction theta_0. After a reduction through invariance arguments, we derive local asymptotic normality (LAN) results in a general high-dimensional framework where the dimension p_n goes to infinity at an arbitrary rate with the sample size n, and where the concentration kappa_n behaves in a completely free way with n, which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones. We identify seven asymptotic regimes, depending on the convergence/divergence properties of (kappa_n), that yield different contiguity rates and different limiting experiments. In each regime, we derive Le Cam optimal tests under specified kappa_n and we compute, from the Le Cam third lemma, asymptotic powers of the classical Watson test under contiguous alternatives. We further establish LAN results with respect to both spike direction and concentration, which allows us to discuss optimality also under unspecified kappa_n. To obtain a full understanding of the non-null behavior of the Watson test, we use martingale CLTs to derive its local asymptotic powers in the broader, semiparametric, model of rotationally symmetric distributions. A Monte Carlo study shows that the finite-sample behaviors of the various tests remarkably agree with our asymptotic results.

Suggested Citation

  • Davy Paindaveine & Thomas Verdebout, 2017. "Detecting the Direction of a Signal on High-dimensional Spheres: Non-null and Le Cam Optimality Results," Working Papers ECARES ECARES 2017-40, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/260378

    Download full text from publisher

    File URL:
    File Function: Full text for the whole work, or for a work part
    Download Restriction: no

    References listed on IDEAS

    1. Christine Cutting & Davy Paindaveine & Thomas Verdebout, 2015. "Testing Uniformity on High-Dimensional Spheres against Contiguous Rotationally Symmetric Alternatives," Working Papers ECARES ECARES 2015-04, ULB -- Universite Libre de Bruxelles.
    2. Hornik, Kurt & GrĂ¼n, 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).
    3. Chikuse, Yasuko, 1991. "High dimensional limit theorems and matrix decompositions on the Stiefel manifold," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 145-162, February.
    4. Davy Paindaveine & Thomas Verdebout, 2013. "Optimal Rank-Based Tests for the Location Parameter of a Rotationally Symmetric Distribution on the Hypersphere," Working Papers ECARES ECARES 2013-36, ULB -- Universite Libre de Bruxelles.
    5. Ley, Christophe & Paindaveine, Davy & Verdebout, Thomas, 2015. "High-dimensional tests for spherical location and spiked covariance," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 79-91.
    6. T. D. Downs, 2003. "Spherical regression," Biometrika, Biometrika Trust, vol. 90(3), pages 655-668, September.
    7. P. V. Larsen, 2002. "Improved likelihood ratio tests on the von Mises--Fisher distribution," Biometrika, Biometrika Trust, vol. 89(4), pages 947-951, December.
    8. R. Arnold & P. E. Jupp, 2013. "Statistics of orthogonal axial frames," Biometrika, Biometrika Trust, vol. 100(3), pages 571-586.
    Full references (including those not matched with items on IDEAS)

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:2013/260378. 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.