The Generalized von Mises℃Fisher Distribution

  • Riccardo Gatto


In this chapter we introduce the broad class of generalized von Mises–Fisher (GvMF) distributions on the unit hypersphere S p − 1, which arises from a generalization of the von Mises–Fisher distribution. This class of distributions has some important information theoretic properties. It is shown that, under constraints on some moments along some fixed directions, and using Kullback–Leibler information as measure, the closest distribution to any predetermined one on S p − 1, has the GvMF form. Lower bounds for the Kullback–Leibler information in this context are also provided. Several connections between GvMF and other directional or linear distributions are given. GvMF distributions can be re-expressed in terms of generalized von Mises distributions when p = 2. GvMF distributions arise as offset multivariate normal distributions, as offset singular normal distributions and as offset distributions from an exponential spherical type of distributions. Various forms of GvMF densities which feature uni- and multimodality, with different shape of modes, girdle and with other particularities are graphically illustrated.


Multivariate Normal Distribution Fisher Distribution Directional Density Leibler Information Unit Hypersphere 
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The author is grateful to Professor Sreenivasa Rao Jammalamadaka, for many constructive and stimulating discussions in the field of directional statistics, to the Editors and to the Referees.


  1. 1.
    Abramowitz M, Stegun IE (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing. Dover, New York. Originally published by the National Bureau of Standards, USA, 10th printingGoogle Scholar
  2. 2.
    Beran R (1979) Exponential models for directional data. Ann Stat 7:1162–1178MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    de Simoni S (1968) Su una estensione dello schema delle curve normali di ordine r alle variabili doppie. Statistica 37:447–474Google Scholar
  4. 4.
    Gatto R (2009) Information theoretic results for circular distributions. Statistics (Ber) 43: 409–421MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gatto R, Jammalamadaka SR (2007) The generalized von Mises distribution. Stat Methodol 4:341–353CrossRefMathSciNetGoogle Scholar
  6. 6.
    Jammalamadaka SR, SenGupta A (2001) Topics in circular statistics. World Scientific, SingaporeMATHCrossRefGoogle Scholar
  7. 7.
    Kullback S (1954) Certain inequalities in information theory and the Cramer–Rao inequality. Ann Math Statist 25:745–751MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Statist 22:79–86MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Maksimov VM (1967) Necessary and sufficient statistics for the family of shifts of probability distributions on continuous bicompact groups. Theoriya Veroyatnostei i ee Primeneniya, 12:307–321 (in Russian); Theory Probab Appl 12:267–280 (English translation)Google Scholar
  10. 10.
    Mardia KV, Jupp PE (2000) Directional statistics. Wiley, ChichesterMATHGoogle Scholar
  11. 11.
    Mardia KV, Kent JT, Bibby JM (1979) Multivariate analysis. Academic, LondonMATHGoogle Scholar
  12. 12.
    Shannon CE (1948) A mathematical theory of communication. Bell Sys Tech J 27:379–423, 623–656MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of BernBernSwitzerland

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