Abstract
Quantiles are recognized tools for risk management and can be seen as minimizers of an \(L^1\)-loss function, but do not define coherent risk measures in general. Expectiles, meanwhile, are minimizers of an \(L^2\)-loss function and define coherent risk measures; they have started to be considered as good alternatives to quantiles in insurance and finance. Quantiles and expectiles belong to the wider family of \(L^p\)-quantiles. We propose here to construct kernel estimators of extreme conditional \(L^p\)-quantiles. We study their asymptotic properties in the context of conditional heavy-tailed distributions, and we show through a simulation study that taking \(p \in (1,2)\) may allow to recover extreme conditional quantiles and expectiles accurately. Our estimators are also showcased on a real insurance data set.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.
Bellini, F., Klar, B., Muller, A., & Gianin, E. R. (2014). Generalized quantiles as risk measures. Insurance: Mathematics and Economics 54, 41–48 (2014).
Cai, J., & Weng, C. (2016). Optimal reinsurance with expectile. Scandinavian Actuarial Journal, 2016(7), 624–645.
Chen, Z. (1996). Conditional \(L_p\)-quantiles and their application to the testing of symmetry in non-parametric regression. Statistics & Probability Letters 29(2), 107–115.
Daouia, A., Gardes, L., & Girard, S. (2013). On kernel smoothing for extremal quantile regression. Bernoulli, 19(5B), 2557–2589.
Daouia, A., Gardes, L., Girard, S., & Lekina, A. (2011). Kernel estimators of extreme level curves. TEST, 20(2), 311–333.
Daouia, A., Girard, S., & Stupfler, G. (2018). Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society: Series B, 80(2), 263–292.
Daouia, A., Girard, S., & Stupfler, G. (2019). Extreme M-quantiles as risk measures: from \(L^1\) to \(L^p\) optimization. Bernoulli, 25(1), 264–309.
Daouia, A., Girard, S., & Stupfler, G. (2020). ExpectHill estimation, extreme risk and heavy tails. Journal of Econometrics, 221(1), 97–117.
Daouia, A., Girard, S., & Stupfler, G. (2020). Tail expectile process and risk assessment. Bernoulli, 26(1), 531–556.
de Haan, L., & Ferreira, A. (2006). Extreme value theory: An introduction. New York: Springer.
El Methni, J., Gardes, L., & Girard, S. (2014). Non-parametric estimation of extreme risk measures from conditional heavy-tailed distributions. Scandinavian Journal of Statistics, 41(4), 988–1012.
El Methni, J., & Stupfler, G. (2017). Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions. Statistica Sinica, 27(2), 907–930.
Embrechts, P., Kluppelberg, C., & Mikosch, T. (1997). Modelling extremal events. Berlin: Springer.
Gardes, L., Girard, S., & Stupfler, G. (2020). Beyond tail median and conditional tail expectation: extreme risk estimation using tail \(L^p-\)optimisation. Scandinavian Journal of Statistics, 47(3), 922–949.
Girard, S., Stupfler, G., & Usseglio-Carleve, A. (2019). An \(L^p\)-quantile methodology for estimating extreme expectiles, preprint. https://hal.inria.fr/hal-02311609v3/document.
Girard, S., Stupfler, G., & Usseglio-Carleve, A. (2021). Nonparametric extreme conditional expectile estimation, To appear in Scandinavian Journal of Statistics. https://hal.archives-ouvertes.fr/hal-02114255.
Jones, M. (1994). Expectiles and M-quantiles are quantiles. Statistics & Probability Letters 20, 149–153.
Koenker, R., & Bassett, G. J. (1978). Regression quantiles. Econometrica, 46(1), 33–50.
Newey, W., & Powell, J. (1987). Asymmetric least squares estimation and testing. Econometrica, 55(4), 819–847.
Ohlsson, E., & Johansson, B. (2010). Non-life insurance pricing with generalized linear models. Berlin: Springer.
Usseglio-Carleve, A. (2018). Estimation of conditional extreme risk measures from heavy-tailed elliptical random vectors. Electronic Journal of Statistics, 12(2), 4057–4093.
Weissman, I. (1978). Estimation of parameters and large quantiles based on the \(k\) largest observations. Journal of the American Statistical Association, 73(364), 812–815.
Acknowledgements
This research was supported by the French National Research Agency under the grant ANR-19-CE40-0013/ExtremReg project. S. Girard gratefully acknowledges the support of the Chair Stress Test, Risk Management and Financial Steering, led by the French Ecole Polytechnique and its Foundation and sponsored by BNP Paribas, and the support of the French National Research Agency in the framework of the Investissements d’Avenir program (ANR-15-IDEX-02).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Girard, S., Stupfler, G., Usseglio-Carleve, A. (2021). Extreme \(L^p\)-quantile Kernel Regression. In: Daouia, A., Ruiz-Gazen, A. (eds) Advances in Contemporary Statistics and Econometrics. Springer, Cham. https://doi.org/10.1007/978-3-030-73249-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-030-73249-3_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-73248-6
Online ISBN: 978-3-030-73249-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)