Estimation of the expected shortfall given an extreme component under conditional extreme value model

Abstract

For two risks, X and Y, the Marginal Expected Shortfall (MES) is defined as \(\mathbb {E}[Y\mid X>F_{X}^{\leftarrow }(1-p)]\), where FX is the distribution function of X and p is small. In this paper we establish consistency and asymptotic normality of an estimator of MES on assuming that (X, Y ) follows a Conditional Extreme Value (CEV) model. The theoretical findings are supported by simulation studies. Our procedure is applied to some financial data.

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References

  1. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics for Extremes: Theory and Applications. Wiley (2004)

  2. Cai, J.-J., Einmahl, J.J.H.J., de Haan, L., Zhou, C.: Estimation of the marginal expected shortfall: the mean when a related variable is extreme. J. R. Stat. Soc. Ser. B 77(2), 417–442 (2015)

    MathSciNet  Article  Google Scholar 

  3. Cai, J.-J.C., Musta, E.: Estimation of the marginal expected shortfall under asymptotic independence. arXiv:1709.04285v1 (2017)

  4. Das, B., Fasen, V.: Risk contagion under regular variation and asymptotic tail independence. arXiv:1603.09406 (2016)

  5. de Haan, L., Ferreira, A.: Extreme value theory. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006)

    Google Scholar 

  6. de Haan, L., Resnick, S.I.: On asymptotic normality of the hill estimator. Communications in Statistics. Stoch. Model. 14, 849–867 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  7. Drees, H., Jannsen, A.: Conditional extreme value models: Fallacies and pitfalls arXiv:1606.02927v1 (2016)

  8. Das, B., Resnick, S.I.: Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17(1), 226–252 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  9. Heffernan, J.E., Resnick, S.I.: Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17(2), 537–571 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  10. Kulik, R., Soulier, P.: The tail empirical process for long memory stochastic volatility sequences. Stoch. Process. Appl. 121(1), 109–134 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  11. Kulik, R., Soulier, P.: Heavy tailed time series with extremal independence. Extremes 18, 273–299 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  12. Resnick, S.I.: Heavy-Tail Phenomena. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). Probabilistic and statistical modeling

    Google Scholar 

  13. Rootzén, H.: Weak convergence of the tail empirical process for dependent sequences. Stoch. Proc. Appl. 119(2), 468–490 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  14. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank all referees for their constructive comments.

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Correspondence to Rafał Kulik.

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Kulik, R., Tong, Z. Estimation of the expected shortfall given an extreme component under conditional extreme value model. Extremes 22, 29–70 (2019). https://doi.org/10.1007/s10687-018-0333-9

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Keywords

  • Marginal expected shortfall
  • Conditional extreme values
  • Regular variation

AMS 2000 Subject Classification

  • 62G32