Robust bounds in multivariate extremes

Sebastian Engelke; Jevgenijs Ivanovs

doi:10.1214/17-AAP1294

December 2017 Robust bounds in multivariate extremes

Sebastian Engelke, Jevgenijs Ivanovs

Ann. Appl. Probab. 27(6): 3706-3734 (December 2017). DOI: 10.1214/17-AAP1294

Abstract

Extreme value theory provides an asymptotically justified framework for estimation of exceedance probabilities in regions where few or no observations are available. For multivariate tail estimation, the strength of extremal dependence is crucial and it is typically modeled by a parametric family of spectral distributions. In this work, we provide asymptotic bounds on exceedance probabilities that are robust against misspecification of the extremal dependence model. They arise from optimizing the statistic of interest over all dependence models within some neighborhood of the reference model. A certain relaxation of these bounds yields surprisingly simple and explicit expressions, which we propose to use in applications. We show the effectiveness of the robust approach compared to classical confidence bounds when the model is misspecified. The results are further applied to quantify the effect of model uncertainty on the Value-at-Risk of a financial portfolio.

References

1.

[1] Ahmadi-Javid, A. (2012). Entropic value-at-risk: A new coherent risk measure. J. Optim. Theory Appl. 155 1105–1123.[1] Ahmadi-Javid, A. (2012). Entropic value-at-risk: A new coherent risk measure. J. Optim. Theory Appl. 155 1105–1123.

2.

[2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228. MR1850791 10.1111/1467-9965.00068[2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228. MR1850791 10.1111/1467-9965.00068

3.

[3] Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability 57. Springer, New York.[3] Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability 57. Springer, New York.

4.

[4] Blanchet, J. and Murthy, K. R. (2016). On distributionally robust extreme value analysis. Preprint. Available at arXiv:1601.06858. 1601.06858[4] Blanchet, J. and Murthy, K. R. (2016). On distributionally robust extreme value analysis. Preprint. Available at arXiv:1601.06858. 1601.06858

5.

[5] Boldi, M.-O. and Davison, A. C. (2007). A mixture model for multivariate extremes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 217–229.[5] Boldi, M.-O. and Davison, A. C. (2007). A mixture model for multivariate extremes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 217–229.

6.

[6] Breuer, T. and Csiszár, I. (2013). Systematic stress tests with entropic plausibility constraints. Journal of Banking & Finance 37 1552–1559.[6] Breuer, T. and Csiszár, I. (2013). Systematic stress tests with entropic plausibility constraints. Journal of Banking & Finance 37 1552–1559.

7.

[7] Breuer, T. and Csiszár, I. (2016). Measuring distribution model risk. Math. Finance 26 395–411.[7] Breuer, T. and Csiszár, I. (2016). Measuring distribution model risk. Math. Finance 26 395–411.

8.

[8] Bücher, A., Dette, H. and Volgushev, S. (2011). New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Statist. 39 1963–2006.[8] Bücher, A., Dette, H. and Volgushev, S. (2011). New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Statist. 39 1963–2006.

9.

[9] Cooley, D., Davis, R. A. and Naveau, P. (2010). The pairwise beta distribution: A flexible parametric multivariate model for extremes. J. Multivariate Anal. 101 2103–2117. MR2671204 10.1016/j.jmva.2010.04.007[9] Cooley, D., Davis, R. A. and Naveau, P. (2010). The pairwise beta distribution: A flexible parametric multivariate model for extremes. J. Multivariate Anal. 101 2103–2117. MR2671204 10.1016/j.jmva.2010.04.007

10.

[10] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.[10] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.

11.

[11] Dey, S. and Juneja, S. (2012). Incorporating fat tails in financial models using entropic divergence measures. Preprint. Available at arXiv:1203.0643. 1203.0643[11] Dey, S. and Juneja, S. (2012). Incorporating fat tails in financial models using entropic divergence measures. Preprint. Available at arXiv:1203.0643. 1203.0643

12.

[12] Dombry, C., Engelke, S. and Oesting, M. (2016). Exact simulation of max-stable processes. Biometrika 103 303–317.[12] Dombry, C., Engelke, S. and Oesting, M. (2016). Exact simulation of max-stable processes. Biometrika 103 303–317.

13.

[13] Einmahl, J. H. J. and Segers, J. (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Statist. 37 2953–2989.[13] Einmahl, J. H. J. and Segers, J. (2009). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Statist. 37 2953–2989.

14.

[14] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.[14] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.

15.

[15] Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). Model uncertainty and var aggregation. Journal of Banking & Finance 37 2750–2764.[15] Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). Model uncertainty and var aggregation. Journal of Banking & Finance 37 2750–2764.

16.

[16] Embrechts, P., Wang, B. and Wang, R. (2015). Aggregation-robustness and model uncertainty of regulatory risk measures. Finance Stoch. 19 763–790.[16] Embrechts, P., Wang, B. and Wang, R. (2015). Aggregation-robustness and model uncertainty of regulatory risk measures. Finance Stoch. 19 763–790.

17.

[17] Glasserman, P. and Xu, X. (2014). Robust risk measurement and model risk. Quant. Finance 14 29–58.[17] Glasserman, P. and Xu, X. (2014). Robust risk measurement and model risk. Quant. Finance 14 29–58.

18.

[18] Hansen, L. P. and Sargent, T. J. (2001). Robust control and model uncertainty. The American Economic Review 91 60–66.[18] Hansen, L. P. and Sargent, T. J. (2001). Robust control and model uncertainty. The American Economic Review 91 60–66.

19.

[19] Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statist. Probab. Lett. 7 283–286.[19] Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statist. Probab. Lett. 7 283–286.

20.

[20] Klüppelberg, C. and May, A. (2006). Bivariate extreme value distributions based on polynomial dependence functions. Math. Methods Appl. Sci. 29 1467–1480.[20] Klüppelberg, C. and May, A. (2006). Bivariate extreme value distributions based on polynomial dependence functions. Math. Methods Appl. Sci. 29 1467–1480.

21.

[21] Mainik, G. and Embrechts, P. (2013). Diversification in heavy-tailed portfolios: Properties and pitfalls. Annals of Actuarial Science 7 26–45.[21] Mainik, G. and Embrechts, P. (2013). Diversification in heavy-tailed portfolios: Properties and pitfalls. Annals of Actuarial Science 7 26–45.

22.

[22] Mainik, G. and Rüschendorf, L. (2010). On optimal portfolio diversification with respect to extreme risks. Finance Stoch. 14 593–623.[22] Mainik, G. and Rüschendorf, L. (2010). On optimal portfolio diversification with respect to extreme risks. Finance Stoch. 14 593–623.

23.

[23] Mitter, S. K. (2008). Convex optimization in infinite dimensional spaces. In Recent Advances in Learning and Control. Lect. Notes Control Inf. Sci. 371 161–179. Springer, London.[23] Mitter, S. K. (2008). Convex optimization in infinite dimensional spaces. In Recent Advances in Learning and Control. Lect. Notes Control Inf. Sci. 371 161–179. Springer, London.

24.

[24] Ouellette, D. V. (1981). Schur complements and statistics. Linear Algebra Appl. 36 187–295.[24] Ouellette, D. V. (1981). Schur complements and statistics. Linear Algebra Appl. 36 187–295.

25.

[25] Pickands, J. III (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981) 49 859–878, 894–902.[25] Pickands, J. III (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981) 49 859–878, 894–902.

26.

[26] Póczos, B. and Schneider, J. G. (2011). On the estimation of $\alpha$-divergences. In AISTATS 609–617.[26] Póczos, B. and Schneider, J. G. (2011). On the estimation of $\alpha$-divergences. In AISTATS 609–617.

27.

[27] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.[27] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.

28.

[28] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York.[28] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York.

29.

[29] Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90 139–156. MR1966556 10.1093/biomet/90.1.139[29] Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90 139–156. MR1966556 10.1093/biomet/90.1.139

30.

[30] Schur, J. (1917). Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math. 147 205–232.[30] Schur, J. (1917). Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. Reine Angew. Math. 147 205–232.

31.

[31] Stephenson, G. (2002). evd: Extreme value distributions. R News 2.[31] Stephenson, G. (2002). evd: Extreme value distributions. R News 2.

32.

[32] Tawn, J. A. (1990). Modelling multivariate extreme value distributions. Biometrika 77 245–253.[32] Tawn, J. A. (1990). Modelling multivariate extreme value distributions. Biometrika 77 245–253.

33.

[33] Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press, Cambridge.[33] Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press, Cambridge.

34.

[34] Zhou, C. (2010). Dependence structure of risk factors and diversification effects. Insurance Math. Econom. 46 531–540.[34] Zhou, C. (2010). Dependence structure of risk factors and diversification effects. Insurance Math. Econom. 46 531–540.

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Sebastian Engelke and Jevgenijs Ivanovs "Robust bounds in multivariate extremes," The Annals of Applied Probability 27(6), 3706-3734, (December 2017). https://doi.org/10.1214/17-AAP1294

Received: 1 July 2016; Published: December 2017

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