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Gardes, Laurent; Girard, Stéphane; Stupfler, Gilles

Beyond tail median and conditional tail expectation: extreme risk estimation using tail \(L^p\)-optimization. (English) Zbl 1454.62143

Scand. J. Stat. 47, No. 3, 922-949 (2020).
Summary: The conditional tail expectation (CTE) is an indicator of tail behavior that takes into account both the frequency and magnitude of a tail event. However, the asymptotic normality of its empirical estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in actuarial and financial applications. A valuable alternative is the median shortfall (MS), although it only gives information about the frequency of a tail event. We construct a class of tail \(L^p\)-medians encompassing the MS and CTE. For \(p\) in \((1,2)\), a tail \(L^p\)-median depends on both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator and another technique to extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on real fire insurance data.

Cited in 5 Documents

MSC:

62G32 Statistics of extreme values; tail inference
60F17 Functional limit theorems; invariance principles
62P20 Applications of statistics to economics

Keywords:

\(L^p\)-optimization; asymptotic normality; conditional tail expectation; extreme value statistics; heavy-tailed distribution; median shortfall
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[1] Acerbi, C., & Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking & Finance, 26, 1487-1503.
[2] Artzner, P., Delbaen, F., Eber, J.‐M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203-228. · Zbl 0980.91042
[3] Basel Committee on Banking Supervision (2013). Fundamental review of the trading book: A revised market risk framework.
[4] Beirlant, J., Goegebeur, Y., Segers, J., & Teugels, J. (2004). Statistics of extremes: Theory and applications. Chichester, UK: John Wiley & Sons. · Zbl 1070.62036
[5] Bingham, N. H., Goldie, C. M., & Teugels, J. L. (1987). Regular variation. Cambridge, UK: Cambridge University Press. · Zbl 0617.26001
[6] Borisov, I. S., & Baklanov, E. A. (2001). Probability inequalities for generalized L‐statistics. Siberian Mathematical Journal, 42, 217-231. · Zbl 0986.60014
[7] Brazauskas, V., Jones, B. L., Puri, M. L., & Zitikis, R. (2008). Estimating conditional tail expectation with actuarial applications in view. Journal of Statistical Planning and Inference, 138, 3590-3604. · Zbl 1152.62027
[8] Cai, J.‐J., Einmahl, J. H. J., deHaan, L., & Zhou, C. (2015). Estimation of the marginal expected shortfall: The mean when a related variable is extreme. Journal of the Royal Statistical Society: Series B, 77, 417-442. · Zbl 1414.91433
[9] Chen, Z. (1996). Conditional L_p‐quantiles and their application to testing of symmetry in non‐parametric regression. Statistics & Probability Letters, 29, 107-115. · Zbl 0865.62021
[10] Daníelsson, J., Jorgensen, B. N., Samorodnitsky, G., Sarma, M., & deVries, C. G. (2013). Fat tails, VaR and subadditivity. Journal of Econometrics, 172, 283-291. · Zbl 1443.62343
[11] Daouia, A., Girard, S., & Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80, 263-292. · Zbl 1383.62235
[12] Daouia, A., Girard, S., & Stupfler, G. (2019). Extreme M‐quantiles as risk measures: From L^1 to L^p optimization. Bernoulli, 25, 264-309. · Zbl 1442.62106
[13] deHaan, L., & Ferreira, A. (2006). Extreme value theory: An introduction. New York, NY: Springer. · Zbl 1101.62002
[14] El Methni, J., Gardes, L., & Girard, S. (2014). Nonparametric estimation of extreme risks from conditional heavy‐tailed distributions. Scandinavian Journal of Statistics, 41, 988-1012. · Zbl 1305.62199
[15] El Methni, J., Gardes, L., & Girard, S. (2018). Kernel estimation of extreme regression risk measures. Electronic Journal of Statistics, 12, 359-398. · Zbl 1388.62141
[16] El Methni, J., & Stupfler, G. (2017). Extreme versions of Wang risk measures and their estimation for heavy‐tailed distributions. Statistica Sinica, 27, 907-930. · Zbl 1362.62183
[17] El Methni, J., & Stupfler, G. (2018). Improved estimators of extreme Wang distortion risk measures for very heavy‐tailed distributions. Econometrics and Statistics, 6, 129-148.
[18] Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events for insurance and finance. Berlin, Heidelberg: Springer. · Zbl 0873.62116
[19] Embrechts, P., Lambrigger, D. P., & Wüthrich, M. V. (2009). Multivariate extremes and the aggregation of dependent risks: Examples and counter‐examples. Extremes, 12, 107-127. · Zbl 1224.91057
[20] Emmer, S., Kratz, M., & Tasche, D. (2015). What is the best risk measure in practice? A comparison of standard measures. Journal of Risk, 18, 31-60.
[21] Gardes, L., & Stupfler, G. (2015). Estimating extreme quantiles under random truncation. TEST, 24, 207-227. · Zbl 1328.62206
[22] Geyer, C. J.. (1996). On the asymptotics of convex stochastic optimization. Unpublished manuscript.
[23] Gneiting, T. (2011). Making and evaluating point forecasts. Journal of the American Statistical Association, 106, 746-762. · Zbl 1232.62028
[24] Gong, J., Li, Y., Peng, L., & Yao, Q. (2015). Estimation of extreme quantiles for functions of dependent random variables. Journal of the Royal Statistical Society: Series B, 77, 1001-1024. · Zbl 1414.62160
[25] Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Annals of Statistics, 3, 1163-1174. · Zbl 0323.62033
[26] Hill, J. B. (2013). Least tail‐trimmed squares for infinite variance autoregressions. Journal of Time Series Analysis, 34, 168-186. · Zbl 1273.62064
[27] Hill, J. B. (2015). Expected shortfall estimation and Gaussian inference for infinite variance time series. Journal of Financial Econometrics, 13, 1-44.
[28] Hua, L., & Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks. Insurance: Mathematics & Economics, 49, 537-546. · Zbl 1228.91039
[29] International Monetary Fund. (2014). Canada: Financial sector assessment program-insurance core principles-detailed assessment of observance. Washington, DC: International Monetary Fund.
[30] Knight, K. (1998). Limiting distributions for L_1 regression estimators under general conditions. Annals of Statistics, 26, 755-770. · Zbl 0929.62021
[31] Knight, K. (1999). Epi‐convergence in distribution and stochastic equi‐semicontinuity (Technical Report). University of Toronto. Retrieved from http://www.utstat.toronto.edu/keith/EPFL/equi.ps.
[32] Koenker, R. (2005). Quantile regression. Cambridge, England: Cambridge University Press. · Zbl 1111.62037
[33] Koenker, R., & Bassett, G. S. (1978). Regression quantiles. Econometrica, 46, 33-50. · Zbl 0373.62038
[34] Kou, S., & Peng, X. (2016). On the measurement of economic tail risk. Operations Research, 64, 1056-1072. · Zbl 1378.91101
[35] Kou, S., Peng, X., & Heyde, C. C. (2013). External risk measures and Basel accords. Mathematics of Operations Research, 38, 393-417. · Zbl 1297.91089
[36] Ling, S. (2005). Self‐weighted least absolute deviation estimation for infinite variance autoregressive models. Journal of the Royal Statistical Society: Series B, 67, 381-393. · Zbl 1069.62069
[37] Newey, W. K., & Powell, J. L. (1987). Asymmetric least squares estimation and testing. Econometrica, 55, 819-847. · Zbl 0625.62047
[38] Organisation for Economic Co‐operation and Development. (2016). Life annuity products and their guarantees. Paris, France: OECD Publishing.
[39] Resnick, S. (2007). Heavy‐tail phenomena: Probabilistic and statistical modeling. New York, NY: Springer. · Zbl 1152.62029
[40] Rockafellar, R. T., & Uryasev, S. (2002). Conditional value‐at‐risk for general loss distributions. Journal of Banking & Finance, 26, 1443-1471.
[41] Tasche, D. (2002). Expected shortfall and beyond. Journal of Banking & Finance, 26, 1519-1533. · Zbl 1119.62106
[42] Tasche, D. (2008). Capital allocation to business units and sub‐portfolios: The Euler principle. In A.Resti (ed.) (Ed.), Pillar II in the new Basel accord: The challenge of economic capital (pp. 423-453). London, England: Risk Books.
[43] Weissman, I. (1978). Estimation of parameters and large quantiles based on the k largest observations. Journal of the American Statistical Association, 73, 812-815. · Zbl 0397.62034
[44] Wüthrich, M., & Merz, M. (2013). Financial modeling, actuarial valuation and solvency in insurance. Berlin, Germany: Springer‐Verlag. · Zbl 1268.91003
[45] Yang, F. (2015). First‐ and second‐order asymptotics for the tail distortion risk measure of extreme risks. Communications in Statistics—Theory and Methods, 44, 520-532. · Zbl 1310.91079
[46] Zhu, L., & Li, H. (2012). Tail distortion risk and its asymptotic analysis. Insurance: Mathematics & Economics, 51, 115-121. · Zbl 1284.91283
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