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