Electronic Journal of Statistics

Estimation of conditional extreme risk measures from heavy-tailed elliptical random vectors

Antoine Usseglio-Carleve

Full-text: Open access

Abstract

In this work, we focus on some conditional extreme risk measures estimation for elliptical random vectors. In a previous paper, we proposed a methodology to approximate extreme quantiles, based on two extremal parameters. We thus propose some estimators for these parameters, and study their consistency and asymptotic normality in the case of heavy-tailed distributions. Thereafter, from these parameters, we construct extreme conditional quantiles estimators, and give some conditions that ensure consistency and asymptotic normality. Using recent results on the asymptotic relationship between quantiles and other risk measures, we deduce estimators for extreme conditional $L_{p}-$quantiles and Haezendonck-Goovaerts risk measures. Under similar conditions, consistency and asymptotic normality are provided. In order to test the effectiveness of our estimators, we propose a simulation study. A financial data example is also proposed.

Article information

Source
Electron. J. Statist., Volume 12, Number 2 (2018), 4057-4093.

Dates
Received: July 2018
First available in Project Euclid: 12 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1544583931

Digital Object Identifier
doi:10.1214/18-EJS1499

Mathematical Reviews number (MathSciNet)
MR3887178

Zentralblatt MATH identifier
07003237

Subjects
Primary: 62H12: Estimation 60E05: Distributions: general theory
Secondary: 62G32: Statistics of extreme values; tail inference

Keywords
Elliptical distribution extreme quantiles extreme value theory Haezendonck-Goovaerts risk measures heavy-tailed distributions $L_{p}-$quantiles

Rights
Creative Commons Attribution 4.0 International License.

Citation

Usseglio-Carleve, Antoine. Estimation of conditional extreme risk measures from heavy-tailed elliptical random vectors. Electron. J. Statist. 12 (2018), no. 2, 4057--4093. doi:10.1214/18-EJS1499. https://projecteuclid.org/euclid.ejs/1544583931


Export citation

References

  • [1] Abanto-Valle, C. A., Bandyopadhyay, D., Lachos, V. H., and Enriquez, I. (2010). Robust bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions., Computational Statistics $\&$ Data Analysis, 54(12):2883–2898.
  • [2] Abramowitz, M., Stegun, I. A., et al. (1966). Handbook of mathematical functions., Applied mathematics series, 55(62):39.
  • [3] Artzner, P., Delbaen, F., Eber, J., and Heath, D. (1999). Coherent measures of risk., Mathematical Finance, 9(3):203–228.
  • [4] Bargès, M., Cossette, H., and Marceau, E. (2009). TVaR-based capital allocation with copulas., Insurance: Mathematics and Economics, 45(3):348–361.
  • [5] Bellini, F., Klar, B., Muller, A., and Gianin, E. R. (2014). Generalized quantiles as risk measures., Insurance: Mathematics and Economics, 54:41–48.
  • [6] Bellini, F. and Rosazza Gianin, E. (2008). On Haezendonck risk measures., Journal of Banking & Finance, 32:986–994.
  • [7] Bernardi, M., Bignozzi, V., and Petrella, L. (2017). On the Lp-quantiles for the Student t distribution., Statistics $\&$ Probability Letters, 128:77–83.
  • [8] Breckling, J. and Chambers, R. (1988). M-quantiles., Biometrika, 75(4):761–771.
  • [9] Cai, J. and Weng, C. (2016). Optimal reinsurance with expectile., Scandinavian Actuarial Journal, 2016(7):624–645.
  • [10] Cambanis, S., Huang, S., and Simons, G. (1981). On the theory of elliptically contoured distributions., Journal of Multivariate Analysis, 11:368–385.
  • [11] Chen, Z. (1996). Conditional Lp-quantiles and their application to the testing of symmetry in non-parametric regression., Statistics $\&$ Probability Letters, 29(2):107–115.
  • [12] Cressie, N. (1988). Spatial prediction and ordinary kriging., Mathematical Geology, 20(4):405–421.
  • [13] Daouia, A., Girard, S., and Stupfler, G. (2017a). Estimation of tail risk based on Extreme Expectiles., Journal of the Royal Statistical Society: Series B, 80(2):263–292.
  • [14] Daouia, A., Girard, S., and Stupfler, G. (2017b). Extreme M-quantiles as risk measures: from L1 to Lp optimization., Bernoulli.
  • [15] de Haan, L. and Ferreira, A. (2006)., Extreme value theory: an introduction. Springer Science & Business Media.
  • [16] de Haan, L. and Resnick, S. (1998). On asymptotic normality of the hill estimator., Communications in Statistics, 14(4):849–866.
  • [17] de Haan, L. and Rootzén, H. (1993). On the estimation of high quantiles., Journal of Statistical Planning and Inference, 35(1):1–13.
  • [18] de Valk, C. (2016). Approximation of high quantiles from intermediate quantiles., Extremes, 19:661–686.
  • [19] Djurčić, D. and Torgašev, A. (2001). Strong asymptotic equivalence and inversion of functions in the class $kc$., Journal of Mathematical Analysis and Applications, 255:383–390.
  • [20] El Methni, J., Gardes, L., Girard, S., and Guillou, A. (2012). Estimation of extreme quantiles from heavy and light tailed distributions., Journal of Statistical Planning and Inference, 142(10):2735–2747.
  • [21] Fang, K.-T., Kotz, S., and Ng, K. W. (1990)., Symmetric multivariate and related distributions. Chapman and Hall.
  • [22] Fielitz, B. D. and Rozelle, J. P. (1983). Stable distributions and the mixtures of distributions hypotheses for common stock returns., Journal of the American Statistical Association, 78(381):28–36.
  • [23] Frahm, G. (2004)., Generalized Elliptical Distributions: Theory and Applications. PhD thesis, Universität zu Köln.
  • [24] Gardes, L. and Girard, S. (2005). Estimating extreme quantiles of weibull tail distributions., Communications in Statistics-Theory and Methods, 34:1065–1080.
  • [25] Gong, J., Li, Y., Peng, L., and Yao, Q. (2015). Estimation of extreme quantiles for functions of dependent random variables., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(5):1001–1024.
  • [26] Goovaerts, M., Kaas, R., Dhaene, J., and Tang, Q. (2004). Some new classes of consistent risk measures., Insurance: Mathematics and Economics, 34:505–516.
  • [27] Haezendonck, J. and Goovaerts, M. (1982). A new premium calculation principle based on Orlicz norms., Insurance: Mathematics and Economics, 1:41–53.
  • [28] Hashorva, E. (2007a). Extremes of conditioned elliptical random vectors., Journal of Multivariate Analysis, 98(8):1583–1591.
  • [29] Hashorva, E. (2007b). Sample extremes of $l_p$-norm asymptotically spherical distributions., Albanian Journal of Mathematics, 1(3):157–172.
  • [30] Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution., The Annals of Statistics, 3(5):1163–1174.
  • [31] Hsing, T. (1991). On Tail Index Estimation Using Dependant Data., The Annals of Statistics, 19(3):1547–1569.
  • [32] Hua, L. and Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks., Insurance: Mathematics and Economics, 49:537–546.
  • [33] Huang, S. T. and Cambanis, S. (1979). Spherically invariant processes: Their nonlinear structure, discrimination, and estimation., Journal of Multivariate Analysis, 9(1):59–83.
  • [34] Jessen, H. A. and Mikosch, T. (2006). Regularly varying functions., Publications de l’Institut Mathematique, 80(94):171–192.
  • [35] Johnson, M. (1987)., Multivariate Statistical Simulation. Wiley $\&$ Sons.
  • [36] Kano, Y. (1994). Consistency property of elliptical probability density functions., Journal of Multivariate Analysis, 51:139–147.
  • [37] Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale parameter generalization., Sankhya: The Indian Journal of Statistics, Series A, 32(4):419–430.
  • [38] Koenker, R. (1992). When are expectiles percentiles?, Econometric Theory, 8:423–424.
  • [39] Koenker, R. and Bassett, G. J. (1978). Regression quantiles., Econometrica, 46(1):33–50.
  • [40] Kratz, M. and Resnick, S. I. (1996). The qq-estimator and heavy tails., Stochastic Models, 12(4):699–724.
  • [41] Kuroda, H. (2013). Quantitative and qualitative monetary easing. In, Speech at a Meeting Held by the Yomiuri International Economic Society in Tokyo, volume 12.
  • [42] Li, Q. and Racine, J. S. (2007)., Nonparametric econometrics: theory and practice. Princeton University Press.
  • [43] Linsmeier, T. J. and Pearson, N. D. (2000). Value at risk., Financial Analysts Journal, 56(2):47–67.
  • [44] Mao, T. and Hu, T. (2012). Second-order properties of the haezendonck-goovaerts risk measure for extreme risks., Insurance: Mathematics and Economics, 51:333–343.
  • [45] Maume-Deschamps, V., Rullière, D., and Usseglio-Carleve, A. (2017). Quantile predictions for elliptical random fields., Journal of Multivariate Analysis, 159:1–17.
  • [46] Maume-Deschamps, V., Rullière, D., and Usseglio-Carleve, A. (2018). Spatial expectile predictions for elliptical random fields., Methodology and Computing in Applied Probability, 20(2):643–671.
  • [47] McNeil, A. J., Frey, R., and Embrechts, P. (2015)., Quantitative risk management: Concepts, techniques and tools. Princeton university press.
  • [48] Newey, W. and Powell, J. (1987). Asymmetric least squares estimation and testing., Econometrica, 55(4):819–847.
  • [49] Opitz, T. (2016). Modeling asymptotically independent spatial extremes based on Laplace random fields., Spatial Statistics, 16:1–18.
  • [50] Owen, J. and Rabinovitch, R. (1983). On the class of elliptical distributions and their applications to the theory of portfolio choice., The Journal of Finance, 38(3):745–752.
  • [51] Parzen, E. (1962). On estimation of a probability density function and mode., Annals of Mathematical Statistics, 33(3):1065–1076.
  • [52] Pickands, J. (1975). Statistical inference using extreme order statistics., The Annals of Statistics, 3(1):119–131.
  • [53] Resnick, S. and Stărică, C. (1995). Consistency of Hill’s Estimator for Dependent Data., Journal of Applied Probability, 32(1):139–167.
  • [54] Schultze, J. and Steinebach, J. (1996). On least squares estimates of an exponential tail coefficient., Statistics $\&$ Decisions, 14(3):353–372.
  • [55] Sobotka, F. and Kneib, T. (2012). Geoadditive expectile regression., Computational Statistics $\&$ Data Analysis, 56(4):755–767.
  • [56] Tang, Q. and Yang, F. (2012). On the Haezendonck-Goovaerts risk measure for extreme risks., Insurance: Mathematics and Economics, 50:217–227.
  • [57] Taylor, J. W. (2008). Estimating Value at Risk and Expected Shortfall Using Expectiles., Journal of Financial Econometrics, 6(2):231–252.
  • [58] Wang, H. J., Li, D., and He, X. (2012). Estimation of high conditional quantiles for heavy-tailed distributions., Journal of the American Statistical Association, 107(500):1453–1464.
  • [59] Xiao, Y. and Valdez, E. (2015). A Black-Litterman asset allocation under Elliptical distributions., Quantitative Finance, 15(3):509–519.