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Ordering results for elliptical distributions with applications to risk bounds

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  • Ansari, Jonathan
  • Rüschendorf, Ludger

Abstract

A classical result of Slepian (1962) for the normal distribution and extended by Das Guptas et al. (1972) for elliptical distributions gives one-sided (lower orthant) comparison criteria for the distributions with respect to the (generalized) correlations. Müller and Scarsini (2000) established that the ordering conditions even characterize the stronger supermodular ordering in the normal case. In the present paper, we extend this result to elliptical distributions. We also derive a similar comparison result for the directionally convex ordering of elliptical distributions. As application, we obtain several results on risk bounds in elliptical classes of risk models under restrictions on the correlations or on the partial correlations. Furthermore, we obtain extensions and strengthening of recent results on risk bounds for various classes of partially specified risk factor models with elliptical dependence structure of the individual risks and the common risk factor. The moderate dependence assumptions on this type of models allow flexible applications and, in consequence, are relevant for improved risk bounds in comparison to the marginal based standard bounds.

Suggested Citation

  • Ansari, Jonathan & Rüschendorf, Ludger, 2021. "Ordering results for elliptical distributions with applications to risk bounds," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:jmvana:v:182:y:2021:i:c:s0047259x20302906
    DOI: 10.1016/j.jmva.2020.104709
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    References listed on IDEAS

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    1. Aas, Kjersti & Czado, Claudia & Frigessi, Arnoldo & Bakken, Henrik, 2009. "Pair-copula constructions of multiple dependence," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 182-198, April.
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    4. Chuancun Yin, 2019. "Stochastic Orderings of Multivariate Elliptical Distributions," Papers 1910.07158, arXiv.org, revised Nov 2019.
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    6. Carole Bernard & Ludger Rüschendorf & Steven Vanduffel & Ruodu Wang, 2017. "Risk bounds for factor models," Finance and Stochastics, Springer, vol. 21(3), pages 631-659, July.
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    8. Müller, Alfred & Scarsini, Marco, 2000. "Some Remarks on the Supermodular Order," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 107-119, April.
    9. Ding, Ying & Zhang, Xinsheng, 2004. "Some stochastic orders of Kotz-type distributions," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 389-396, October.
    10. Chuancun Yin, 2019. "Stochastic ordering of Gini indexes for multivariate elliptical random variables," Papers 1908.01943, arXiv.org, revised Sep 2019.
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