Bernoulli

  • Bernoulli
  • Volume 19, Number 5B (2013), 2557-2589.

On kernel smoothing for extremal quantile regression

Abdelaati Daouia, Laurent Gardes, and Stéphane Girard

Full-text: Open access

Abstract

Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory far enough into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Specifically, we elucidate their limit distributions when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their (intermediate or extreme) order with the oscillations of the quantile function and a von-Mises property of the conditional distribution. A simulation experiment and an illustration on real data were presented. The real data are the American electric data where the estimation of conditional extremes is found to be of genuine interest.

Article information

Source
Bernoulli, Volume 19, Number 5B (2013), 2557-2589.

Dates
First available in Project Euclid: 3 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1386078613

Digital Object Identifier
doi:10.3150/12-BEJ466

Mathematical Reviews number (MathSciNet)
MR3160564

Zentralblatt MATH identifier
1281.62097

Keywords
asymptotic normality extreme quantile extreme-value index kernel smoothing regression von-Mises condition

Citation

Daouia, Abdelaati; Gardes, Laurent; Girard, Stéphane. On kernel smoothing for extremal quantile regression. Bernoulli 19 (2013), no. 5B, 2557--2589. doi:10.3150/12-BEJ466. https://projecteuclid.org/euclid.bj/1386078613


Export citation

References

  • [1] Beirlant, J. and Goegebeur, Y. (2004). Local polynomial maximum likelihood estimation for Pareto-type distributions. J. Multivariate Anal. 89 97–118.
  • [2] Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications. Chichester: Wiley.
  • [3] Berlinet, A., Gannoun, A. and Matzner-Løber, E. (2001). Asymptotic normality of convergent estimates of conditional quantiles. Statistics 35 139–169.
  • [4] Chavez-Demoulin, V. and Davison, A.C. (2005). Generalized additive modelling of sample extremes. J. Roy. Statist. Soc. Ser. C 54 207–222.
  • [5] Chernozhukov, V. (2005). Extremal quantile regression. Ann. Statist. 33 806–839.
  • [6] Daouia, A., Florens, J.P. and Simar, L. (2010). Frontier estimation and extreme value theory. Bernoulli 16 1039–1063.
  • [7] Daouia, A., Gardes, L., Girard, S. and Lekina, A. (2011). Kernel estimators of extreme level curves. TEST 20 311–333.
  • [8] Davison, A.C. and Ramesh, N.I. (2000). Local likelihood smoothing of sample extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 191–208.
  • [9] Davison, A.C. and Smith, R.L. (1990). Models for exceedances over high thresholds. J. Roy. Statist. Soc. Ser. B 52 393–442.
  • [10] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. New York: Springer.
  • [11] Drees, H. (1995). Refined Pickands estimators of the extreme value index. Ann. Statist. 23 2059–2080.
  • [12] Feigin, P.D. and Resnick, S.I. (1997). Linear programming estimators and bootstrapping for heavy tailed phenomena. Adv. in Appl. Probab. 29 759–805.
  • [13] Gannoun, A. (1990). Estimation non paramétrique de la médiane conditionnelle, médianogramme et méthode du noyau. Publications de l’Institut de Statistique de l’Université de Paris XXXXVI 11–22.
  • [14] Gardes, L. and Girard, S. (2008). A moving window approach for nonparametric estimation of the conditional tail index. J. Multivariate Anal. 99 2368–2388.
  • [15] 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.
  • [16] Gardes, L., Girard, S. and Lekina, A. (2010). Functional nonparametric estimation of conditional extreme quantiles. J. Multivariate Anal. 101 419–433.
  • [17] Girard, S. and Jacob, P. (2008). Frontier estimation via kernel regression on high power-transformed data. J. Multivariate Anal. 99 403–420.
  • [18] Girard, S. and Menneteau, L. (2005). Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries. J. Statist. Plann. Inference 135 433–460.
  • [19] Greene, W.H. (1990). A gamma-distributed stochastic frontier model. J. Econometrics 46 141–163.
  • [20] Hall, P. and Tajvidi, N. (2000). Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Statist. Sci. 15 153–167.
  • [21] Härdle, W. and Stoker, T.M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84 986–995.
  • [22] Hendricks, W. and Koenker, R. (1992). Hierarchical spline models for conditional quantiles and the demand for electricity. J. Amer. Statist. Assoc. 87 58–68.
  • [23] Jurečková, J. (2007). Remark on extreme regression quantile. Sankhyā 69 87–100.
  • [24] Koenker, R. and Geling, O. (2001). Reappraising medfly longevity: A quantile regression survival analysis. J. Amer. Statist. Assoc. 96 458–468.
  • [25] Park, B.U. (2001). On nonparametric estimation of data edges. J. Korean Statist. Soc. 30 265–280.
  • [26] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065–1076.
  • [27] Ruppert, D., Wand, M.P. and Carroll, R.J. (2003). Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics 12. Cambridge: Cambridge Univ. Press.
  • [28] Samanta, M. (1989). Nonparametric estimation of conditional quantiles. Statist. Probab. Lett. 7 407–412.
  • [29] Smith, R.L. (1989). Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone. Statist. Sci. 4 367–393.
  • [30] Stone, C.J. (1977). Consistent nonparametric regression. Ann. Statist. 5 595–645.
  • [31] Stute, W. (1986). Conditional empirical processes. Ann. Statist. 14 638–647.
  • [32] Tsay, R.S. (2002). Analysis of Financial Time Series. Hoboken, NJ: Wiley.
  • [33] Yu, K. and Jones, M.C. (1998). Local linear quantile regression. J. Amer. Statist. Assoc. 93 228–237.