## Electronic Journal of Statistics

### Kernel estimation of extreme regression risk measures

#### Abstract

The Regression Conditional Tail Moment (RCTM) is the risk measure defined as the moment of order $b\geq0$ of a loss distribution above the upper $\alpha$-quantile where $\alpha\in (0,1)$ and when a covariate information is available. The purpose of this work is first to establish the asymptotic properties of the RCTM in case of extreme losses, i.e when $\alpha\to 0$ is no longer fixed, under general extreme-value conditions on their distribution tail. In particular, no assumption is made on the sign of the associated extreme-value index. Second, the asymptotic normality of a kernel estimator of the RCTM is established, which allows to derive similar results for estimators of related risk measures such as the Regression Conditional Tail Expectation/Variance/Skewness. When the distribution tail is upper bounded, an application to frontier estimation is also proposed. The results are illustrated both on simulated data and on a real dataset in the field of nuclear reactors reliability.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 359-398.

Dates
First available in Project Euclid: 15 February 2018

https://projecteuclid.org/euclid.ejs/1518663657

Digital Object Identifier
doi:10.1214/18-EJS1392

Mathematical Reviews number (MathSciNet)
MR3763910

Zentralblatt MATH identifier
1388.62141

#### Citation

El Methni, Jonathan; Gardes, Laurent; Girard, Stéphane. Kernel estimation of extreme regression risk measures. Electron. J. Statist. 12 (2018), no. 1, 359--398. doi:10.1214/18-EJS1392. https://projecteuclid.org/euclid.ejs/1518663657

#### References

• [1] Aarssen, K., and de Haan, L. (1994). On the maximal life span of humans., Mathematical Population Studies, 4(4), 259–281.
• [2] Aragon, Y., Daouia, A., and Thomas-Agnan, C. (2005). Nonparametric frontier estimation: a conditional quantile-based approach., Journal of Econometric Theory, 21(2), 358–389.
• [3] Artzner, P., Delbaen, F., Eber, J.M., and Heath, D. (1999). Coherent measures of risk., Mathematical Finance, 9, 203–228.
• [4] Beirlant, J., Goegebeur, Y., Segers, J., and Teugels J. (2004)., Statistics of extremes: Theory and applications, Wiley.
• [5] Berlinet, A., Gannoun, A., and Matzner-Lober, E. (2001). Asymptotic normality of convergent estimates of conditional quantiles., Statistics, 35, 139–169.
• [6] Bingham, N.H., Goldie, C.M., and Teugels, J.L. (1987)., Regular Variation, Cambridge University Press.
• [7] Cazals, C., Florens, J.-P., and Simar, L. (2002). Nonparametric frontier estimation: A robust approach., Journal of Econometrics, 106(1), 1–25.
• [8] Cowling, A., and Hall, P. (1996). On pseudodata methods for removing boundary effects in kernel density estimation., Journal of the Royal Statistical Society B, 58, 551–563.
• [9] Daouia, A., Florens, J-P., and Simar, L. (2010). Frontier estimation and extreme value theory., Bernoulli, 16, 1039–1063.
• [10] Daouia, A., Gardes, L., Girard, S. and Lekina, A. (2011). Kernel estimators of extreme level curves., TEST, 20, 311–333.
• [11] Daouia, A., Gardes, L., and Girard, S. (2013). On kernel smoothing for extremal quantile regression., Bernoulli, 19, 2557–2589.
• [12] Daouia, A., Noh, H., and Park, B.U. (2016). Data envelope fitting with constrained polynomial spline., Journal of the Royal Statistical Society, B, 78(1), 3–36.
• [13] Daouia, A., and Simar, L. (2005). Robust nonparametric estimators of monotone boundaries., Journal of Multivariate Analysis, 96, 311–331.
• [14] Deprins, D., Simar, L., and Tulkens, H. (1984). Measuring Labor Efficiency in Post Offices. in, The Performance of Public Enterprises: Concepts and Measurements by M. Marchand, P. Pestieau and H. Tulkens, North Holland ed, Amsterdam.
• [15] Drees, H. (1995). Refined Pickands estimators of the extreme value index., The Annals of Statistics, 23, 2059–2080.
• [16] El Methni, J., Gardes, L., and Girard, S. (2014). Nonparametric estimation of extreme risk measures from conditional heavy-tailed distributions., Scandinavian Journal of Statistics, 41(4), 988–1012.
• [17] Farrel, M.J. (1957). The measurement of productive efficiency., Journal of the Royal Statistical Society A, 120, 253–281.
• [18] Fraga Alves, M.I., Gomes, M.I., de Haan, L., and Neves, C. (2007). A note on second order conditions in extreme value theory: linking general and heavy tail conditions., REVSTAT - Statistical Journal, 5(3), 285–304.
• [19] Fraga Alves, M.I., Neves, C., and Rosário, P. (2017). A general estimator for the right endpoint with an application to supercentenarian women’s records., Extremes, 20, 199–237.
• [20] Gardes, L., and Girard, S. (2010). Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels., Extremes, 13, 177–204.
• [21] Gasser, T., and Müller, H. (1979). Kernel estimation of regression functions. In: Gasser T., Rosenblatt M. (eds), Smoothing techniques for curve estimation, Lecture Notes in Mathematics, vol 757, pp. 23–68, Springer, Berlin Heidelberg.
• [22] Geffroy, J. (1964). Sur un problème d’estimation géométrique., Publications de l’Institut de Statistique de l’Université de Paris, XIII, 191–200.
• [23] Gijbels, I., Mammen, E., Park, B.U., and Simar, L. (1999). On estimation of monotone and concave frontier functions., Journal of the American Statistical Association, 94, 220–228.
• [24] Girard, S., Guillou, A., and Stupfler, G. (2013). Frontier estimation with kernel regression on high order moments., Journal of Multivariate Analysis, 116, 172–189.
• [25] Girard, S., and Jacob, P. (2004). Extreme values and kernel estimates of point processes boundaries., ESAIM: Probability and Statistics, 8, 150–168.
• [26] Girard, S., and Jacob, P. (2008). Frontier estimation via kernel regression on high power transformed data., Journal of Multivariate Analysis, 99, 403–420.
• [27] Girard, S., and Nazin, A.V. (2014). L1-optimal linear programming estimator for periodic frontier functions with Holder continuous derivative., Automation and Remote Control, 75(12), 2152–2169.
• [28] de Haan, L., and Ferreira, A. (2006)., Extreme Value Theory: An introduction, Springer Series in Operations Research and Financial Engineering, Springer.
• [29] Hall, P., Park, B.U., and Stern, S.E. (1998). On polynomial estimators of frontiers and boundaries., Journal of Multivariate Analysis, 66, 71–98.
• [30] Härdle, W., Hall, P., and Simar, L. (1995). Iterated bootstrap with application to frontier models., J. Productivity Anal., 6, 63–76.
• [31] Härdle, W., Park, B.U., and Tsybakov, A.B. (1995). Estimation of a non sharp support boundaries., Journal of Multivariate Analysis, 43, 205–218.
• [32] Hong, J., and Elshahat, A. (2010). Conditional tail variance and conditional tail skewness., Journal of Financial and Economic Practice, 10(1), 147–156.
• [33] Hua, L., and Joe, H. (2011). Second order regular variation and conditional tail expectation of multiple risks., Insurance: Mathematics and Economics, 49(3), 537–546.
• [34] Jacob, P., and Suquet, P. (1995). Estimating the edge of a Poisson process by orthogonal series., Journal of Statistical Planning and Inference, 46, 215–234.
• [35] Jorion, P. (2007)., Value at risk: the new benchmark for managing financial risk, McGraw-Hill New York.
• [36] Kyung-Joon, C., and Schucany, W. (1998). Nonparametric kernel regression estimation near endpoints., Journal of Statistical Planning and Inference, 66, 289–304.
• [37] Knight, K. (2001). Limiting distributions of linear programming estimators., Extremes, 4(2), 87–103.
• [38] Koenker, R., and Geling, O. (2001). Reappraising medfly longevity: A quantile regression survival analysis., Journal of American Statistical Association, 96, 458–468.
• [39] Korostelev, A., Simar, L., and Tsybakov, A.B. (1995). Efficient estimation of monotone boundaries., The Annals of Statistics, 23, 476–489.
• [40] Korostelev, A.P., and Tsybakov, A.B. (1993). Minimax theory of image reconstruction. in, Lecture Notes in Statistics, 82, Springer-Verlag, New York.
• [41] Parzen, E. (1962). On estimation of a probability density function and mode., The Annals of Mathematical Statistics, 33, 1065–1076.
• [42] Resnick, S.I. (2008)., Extreme Values, Regular Variation, and Point Processes, Springer.
• [43] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function., The Annals of Mathematical Statistics, 832–837.
• [44] Samanta, T. (1989). Non-parametric estimation of conditional quantiles., Statistics and Probability Letters, 7, 407–412.
• [45] Smith, R.L. (1989). Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone (with discussion)., Statistical Science, 4, 367–393.
• [46] Stone, C.J. (1977). Consistent nonparametric regression (with discussion)., The Annals of Statistics, 5, 595–645.
• [47] Stute, W. (1986). Conditional empirical processes., The Annals of Statistics, 14, 638–647.
• [48] Tsay, R.S. (2002)., Analysis of financial time series. Wiley, New-York.
• [49] Valdez, E.A. (2005). Tail conditional variance for elliptically contoured distributions., Belgian Actuarial Bulletin, 5, 26–36.