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Extended Generalized Skew-Elliptical Distributions and their Moments

Author

Listed:
  • Zinoviy Landsman

    (University of Haifa)

  • Udi Makov

    (University of Haifa)

  • Tomer Shushi

    (University of Haifa)

Abstract

This paper constructs a family of multivariate distributions which extends the class of generalized skew-elliptical (GSE) distributions, introduced by Azzalini and Capitanio (J. Roy. Statist. Soc. Ser. B, 65, 367–389, 2003), and derives formulae for it’s characteristics; mean and covariance. The extended GSE family is particularly relevant whenever the need arises to model data by skewed distributions, as is the case in actuarial science, risk management and other branches of science. The paper also generalizes the skew-normal distribution in the sense of Azzalini and Dalla Valle (Biometrika, 83, 715–726, 1996). Furthermore, for estimation purposes, the maximum likelihood equations are derived. Finally, a numerical examples is provided which demonstrates that the extended GSE distribution offers a better fit in comparison to the GSE distribution.

Suggested Citation

  • Zinoviy Landsman & Udi Makov & Tomer Shushi, 2017. "Extended Generalized Skew-Elliptical Distributions and their Moments," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(1), pages 76-100, February.
    • Handle: RePEc:spr:sankha:v:79:y:2017:i:1:d:10.1007_s13171-016-0090-2
    DOI: 10.1007/s13171-016-0090-2
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    References listed on IDEAS

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    1. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    2. Reinaldo B. Arellano-Valle & Marc G. Genton, 2010. "Multivariate extended skew-t distributions and related families," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 201-234.
    3. Landsman, Zinoviy, 2006. "On the generalization of Stein's Lemma for elliptical class of distributions," Statistics & Probability Letters, Elsevier, vol. 76(10), pages 1012-1016, May.
    4. Nadarajah, Saralees & Kotz, Samuel, 2003. "Skewed distributions generated by the normal kernel," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 269-277, November.
    5. Marsaglia, George & Marsaglia, John, 2004. "Evaluating the Anderson-Darling Distribution," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 9(i02).
    6. Marc Genton & Nicola Loperfido, 2005. "Generalized skew-elliptical distributions and their quadratic forms," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(2), pages 389-401, June.
    7. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    8. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
    9. 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.
    10. Adelchi Azzalini, 2005. "The Skew‐normal Distribution and Related Multivariate Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 159-188, June.
    11. Barry Arnold & Robert Beaver & A. Azzalini & N. Balakrishnan & A. Bhaumik & D. Dey & C. Cuadras & J. Sarabia & Barry Arnold & Robert Beaver, 2002. "Skewed multivariate models related to hidden truncation and/or selective reporting," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(1), pages 7-54, June.
    12. Landsman, Zinoviy & Neslehová, Johanna, 2008. "Stein's Lemma for elliptical random vectors," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 912-927, May.
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    2. Jorge M. Arevalillo & Hilario Navarro, 2019. "A stochastic ordering based on the canonical transformation of skew-normal vectors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 475-498, June.

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