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On the computation of multivariate scenario sets for the skew-t and generalized hyperbolic families

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  • Giorgi, Emanuele
  • McNeil, Alexander J.

Abstract

The problem of computing multivariate scenarios sets for skewed distributions is motivated by the potential use of such sets in the stress testing of insurance companies and banks. Multivariate scenario sets based on the notion of half-space depth (HD) are considered and the notion of expectile depth (ED) is introduced. These depth concepts facilitate the definition of convex scenario sets, which generalize the concepts of quantiles and expectiles to higher dimensions. In the case of elliptical distributions the scenario sets coincide with the regions encompassed by the contours of the density function. In the context of multivariate skewed distributions, the equivalence of depth contours and density contours does not hold in general. Two parametric families that account for skewness and heavy tails are analysed: the generalized hyperbolic and the skew-t distributions. By making use of a canonical form representation, where skewness is completely absorbed by one component, it is shown that the HD contours of these distributions are near-elliptical; in the case of the skew-Cauchy distribution the HD contours are exactly elliptical. A measure of multivariate skewness as a deviation from angular symmetry is proposed. This measure is shown to explain the quality of the elliptical approximation for the HD contours.

Suggested Citation

  • Giorgi, Emanuele & McNeil, Alexander J., 2016. "On the computation of multivariate scenario sets for the skew-t and generalized hyperbolic families," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 205-220.
    • Handle: RePEc:eee:csdana:v:100:y:2016:i:c:p:205-220
    DOI: 10.1016/j.csda.2014.06.024
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    References listed on IDEAS

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    1. Masse, J. C. & Theodorescu, R., 1994. "Halfplane Trimming for Bivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 48(2), pages 188-202, February.
    2. Behboodian, J. & Jamalizadeh, A. & Balakrishnan, N., 2006. "A new class of skew-Cauchy distributions," Statistics & Probability Letters, Elsevier, vol. 76(14), pages 1488-1493, August.
    3. Adelchi Azzalini & Marc G. Genton, 2008. "Robust Likelihood Methods Based on the Skew‐t and Related Distributions," International Statistical Review, International Statistical Institute, vol. 76(1), pages 106-129, April.
    4. 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.
    5. Thomas Breuer & Martin Jandacka & Klaus Rheinberger & Martin Summer, 2009. "How to Find Plausible, Severe and Useful Stress Scenarios," International Journal of Central Banking, International Journal of Central Banking, vol. 5(3), pages 205-224, September.
    6. 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.
    7. 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.
    8. Nolan, D., 1992. "Asymptotics for multivariate trimming," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 157-169, August.
    9. 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.
    10. Jones, M. C., 1994. "Expectiles and M-quantiles are quantiles," Statistics & Probability Letters, Elsevier, vol. 20(2), pages 149-153, May.
    11. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    12. McNeil, Alexander J. & Smith, Andrew D., 2012. "Multivariate stress scenarios and solvency," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 299-308.
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    Cited by:

    1. Cascos, Ignacio & Ochoa, Maicol, 2021. "Expectile depth: Theory and computation for bivariate datasets," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    2. repec:cte:wsrepe:28434 is not listed on IDEAS

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