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A general setting for symmetric distributions and their relationship to general distributions

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  • Jupp, P.E.
  • Regoli, G.
  • Azzalini, A.

Abstract

A standard method of obtaining non-symmetrical distributions is that of modulating symmetrical distributions by multiplying the densities by a perturbation factor. This has been considered mainly for central symmetry of a Euclidean space in the origin. This paper enlarges the concept of modulation to the general setting of symmetry under the action of a compact topological group on the sample space. The main structural result relates the density of an arbitrary distribution to the density of the corresponding symmetrised distribution. Some general methods for constructing modulating functions are considered. The effect that transformations of the sample space have on symmetry of distributions is investigated. The results are illustrated by general examples, many of them in the setting of directional statistics.

Suggested Citation

  • Jupp, P.E. & Regoli, G. & Azzalini, A., 2016. "A general setting for symmetric distributions and their relationship to general distributions," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 107-119.
    • Handle: RePEc:eee:jmvana:v:148:y:2016:i:c:p:107-119
    DOI: 10.1016/j.jmva.2016.02.011
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    References listed on IDEAS

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    1. Ley, Christophe & Paindaveine, Davy, 2010. "Multivariate skewing mechanisms: A unified perspective based on the transformation approach," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1685-1694, December.
    2. Umbach, Dale & Jammalamadaka, S. Rao, 2009. "Building asymmetry into circular distributions," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 659-663, March.
    3. 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.
    4. Adelchi Azzalini & Giuliana Regoli, 2012. "Some properties of skew-symmetric distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(4), pages 857-879, August.
    5. J. T. Chang & D. Pollard, 1997. "Conditioning as disintegration," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 51(3), pages 287-317, November.
    6. Chikuse, Y. & Jupp, P. E., 2004. "A test of uniformity on shape spaces," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 163-176, January.
    7. R. Arnold & P. E. Jupp, 2013. "Statistics of orthogonal axial frames," Biometrika, Biometrika Trust, vol. 100(3), pages 571-586.
    8. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
    9. Adelchi Azzalini, 2012. "Selection models under generalized symmetry settings," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(4), pages 737-750, August.
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    Cited by:

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    3. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    4. Jose Ameijeiras-Alonso & Christophe Ley & Arthur Pewsey & Thomas Verdebout, 2021. "On optimal tests for circular reflective symmetry about an unknown central direction," Statistical Papers, Springer, vol. 62(4), pages 1651-1674, August.

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