Skip to main content

Maximum likelihood characterization of rotationally symmetric distributions on the sphere

Abstract

A classical characterization result, which can be traced back to Gauss, states that the maximum likelihood estimator (MLE) of the location parameter equals the sample mean for any possible univariate samples of any possible sizes n if and only if the samples are drawn from a Gaussian population. A similar result, in the two-dimensional case, is given in von Mises (1918) for the Fisher-von Mises-Langevin (FVML) distribution, the equivalent of the Gaussian law on the unit circle. Half a century later, Bingham and Mardia (1975) extend the result to FVML distributions on the unit sphere \(\mathcal{S}^{k-1}:=\{{\ensuremath{\mathbf{v}}}\in{\mathbb R}^k:{\ensuremath{\mathbf{v}}}'{\ensuremath{\mathbf{v}}}=1\}\), k ≥ 2. In this paper, we present a general MLE characterization theorem for a large subclass of rotationally symmetric distributions on \(\mathcal{S}^{k-1}\), k ≥ 2, including the FVML distribution.

This is a preview of subscription content, access via your institution.

References

  • Aczél, J. and Dhombres, J. (1989). Functional equations in several variables with applications to mathematics, information theory and to the natural and social sciences. In Encyclopedia Math. Appl., vol. 31. Cambridge University Press, Cambridge.

    Google Scholar 

  • Arnold, K.J. (1941). On spherical probability distributions. Ph.D. thesis, Massachusetts Institute of Technology.

  • Azzalini, A. and Genton, M.G. (2007). On Gauss’s characterization of the normal distribution. Bernoulli, 13, 169–174.

    MathSciNet  MATH  Article  Google Scholar 

  • Bingham, M.S. and Mardia, K.V. (1975). Characterizations and applications. In Statistical Distributions for Scientific Work, vol. 3, (G.P. Patil, S. Kotz and J.K. Ord, eds.). Reidel, Dordrecht and Boston, pp. 387–398.

    Google Scholar 

  • Breitenberger, E. (1963). Analogues of the normal distribution on the circle and the sphere. Biometrika, 50, 81–88.

    MathSciNet  MATH  Google Scholar 

  • Chang, T. (2004). Spatial statistics. Statist. Sci., 19, 624–635.

    MathSciNet  MATH  Article  Google Scholar 

  • Duerinckx, M., Ley, C. and Swan, Y. (2012). Maximum likelihood characterization of distributions. arXiv:1207.2805.

  • Fisher, R.A. (1953). Dispersion on a sphere. Proc. R. Soc. Lond. Ser. A, 217, 295–305.

    MATH  Article  Google Scholar 

  • Fisher, N.I. (1985). Spherical medians. J. R. Stat. Soc. Ser. B, 47, 342–348.

    MATH  Google Scholar 

  • Gauss, C.F. (1809). Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Perthes et Besser, Hamburg. English translation by C.H. Davis, reprinted by Dover, New York (1963).

  • Jones, M.C. and Pewsey, A. (2005). A family of symmetric distributions on the circle. J. Amer. Statist. Assoc., 100, 1422–1428.

    MathSciNet  MATH  Article  Google Scholar 

  • Kagan, A.M., Linnik, Y.V. and Rao, C.R. (1973). Characterization problems in mathematical statistics. Wiley, New York.

    MATH  Google Scholar 

  • Langevin, P. (1905). Sur la théorie du magnétisme. J. Phys., 4, 678–693; Magnétisme et théorie des électrons. Ann. Chim. Phys., 5, 70–127.

    Google Scholar 

  • Ley, C., Swan, Y., Thiam, B. and Verdebout, T. (2013). Optimal R-estimation of a spherical location. Stat. Sinica, 23, to appear.

  • Mardia, K.V. (1972). Statistics of directional data. Academic Press, London.

    MATH  Google Scholar 

  • Mardia, K.V. (1975a). Statistics of directional data. J. R. Stat. Soc. Ser. B, 37, 349–393.

    MathSciNet  MATH  Google Scholar 

  • Mardia, K.V. (1975b). Characterization of directional distributions. In Statistical Distributions for Scientific Work, vol. 3, (G.P. Patil, S. Kotz and J.K. Ord, eds.). Reidel, Dordrecht and Boston, pp. 365–385.

    Google Scholar 

  • Mardia, K.V. and Jupp, P.E. (2000). Directional statistics. Wiley, Chichester.

    MATH  Google Scholar 

  • Purkayastha, S. (1991). A rotationally symmetric directional distribution: obtained through maximum likelihood characterization. Sankhyā Ser. A, 53, 70–83.

    MathSciNet  MATH  Google Scholar 

  • Saw, J.G. (1978). A family of distributions on the m-sphere and some hypothesis tests. Biometrika, 65, 69–73.

    MathSciNet  MATH  Google Scholar 

  • Von Mises, R. (1918). Uber die ‘Ganzzahligkeit’ der Atomgewichte und verwandte Fragen. Phys. Z., 19, 490–500.

    MATH  Google Scholar 

  • Watson, G.S. (1982). Distributions on the circle and sphere. J. Appl. Probab., 19, 265–280.

    Article  Google Scholar 

  • Watson, G.S. (1983). Statistics on spheres. Wiley, New York.

    MATH  Google Scholar 

  • Whittaker, E.T. and Watson, G.N. (1990). A Course in Modern Analysis, 4th edn. Cambridge University Press.

Download references

Acknowledgement

Christophe Ley thanks the Fonds National de la Recherche Scientifique, Communauté française de Belgique, for support via a Mandat de Chargé de Recherche.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christophe Ley.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Duerinckx, M., Ley, C. Maximum likelihood characterization of rotationally symmetric distributions on the sphere. Sankhya A 74, 249–262 (2012). https://doi.org/10.1007/s13171-012-0004-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13171-012-0004-x

Keywords and phrases

  • Cauchy’s functional equation
  • characterization theorem
  • Fisher-von Mises-Langevin distribution
  • maximum likelihood estimator
  • rotationally symmetric distributions on the sphere

AMS (2000) subject classification

  • Primary 62H05
  • 62E10
  • Secondary 60E05