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Relationships between distributions with certain symmetries


  • Jones, M.C.


The genesis of two-way links between the inverse Gaussian and Birnbaum–Saunders distributions is explored and extended. The most general results apply to pairs of distributions with a general ‘S-symmetry’ structure involving a self-inverse function closely related to a transformation function with certain properties. These general results arise by transformation from very simple properties of the familiar Azzalini-type skew-symmetric distributions. They specialise again to relationships between R-symmetric and log-symmetric distributions, between various models related to the inverse Gaussian and Birnbaum–Saunders distributions, relationships involving the sinh–arcsinh transformation, and others. Simple random variate generation is a practical consequence of these relationships.

Suggested Citation

  • Jones, M.C., 2012. "Relationships between distributions with certain symmetries," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1737-1744.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:9:p:1737-1744
    DOI: 10.1016/j.spl.2012.05.014

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    References listed on IDEAS

    1. J. Rosco & M. Jones & Arthur Pewsey, 2011. "Skew t distributions via the sinh-arcsinh transformation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(3), pages 630-652, November.
    2. Ramesh C. Gupta & Debasis Kundu, 2011. "Weighted inverse Gaussian -- a versatile lifetime model," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(12), pages 2695-2708, February.
    3. Leiva, Víctor & Hernández, Hugo & Sanhueza, Antonio, 2008. "An R Package for a General Class of Inverse Gaussian Distributions," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 26(i04).
    4. Antonio Sanhueza & Víctor Leiva & N. Balakrishnan, 2008. "A new class of inverse Gaussian type distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 68(1), pages 31-49, June.
    5. M. C. Jones & Arthur Pewsey, 2009. "Sinh-arcsinh distributions," Biometrika, Biometrika Trust, vol. 96(4), pages 761-780.
    6. M.C. Jones, 2007. "Connecting Distributions with Power Tails on the Real Line, the Half Line and the Interval," International Statistical Review, International Statistical Institute, vol. 75(1), pages 58-69, April.
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    Cited by:

    1. Lemonte, Artur J., 2013. "A new extended Birnbaum–Saunders regression model for lifetime modeling," Computational Statistics & Data Analysis, Elsevier, vol. 64(C), pages 34-50.
    2. Mudholkar, Govind S. & Yu, Ziji & Awadalla, Saria S., 2015. "The mode-centric M-Gaussian distribution: A model for right skewed data," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 1-10.


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