Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Uniform asymptotic properties of a nonparametric regression estimator of conditional tails

Yuri Goegebeur, Armelle Guillou, and Gilles Stupfler

Full-text: Open access

Abstract

We consider a nonparametric regression estimator of conditional tails introduced by Goegebeur, Y., Guillou, A., Schorgen, G. (2013). Nonparametric regression estimation of conditional tails – the random covariate case. It is shown that this estimator is uniformly strongly consistent on compact sets and its rate of convergence is given.

Résumé

Nous considérons l’estimateur à noyau de l’indice des valeurs extrêmes conditionnel présenté dans Goegebeur, Y., Guillou, A., Schorgen, G. (2013). Nonparametric regression estimation of conditional tails – the random covariate case. Nous montrons la consistance uniforme presque sûre de cet estimateur sur les compacts et nous calculons sa vitesse de convergence presque sûre.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 1190-1213.

Dates
Received: 22 November 2013
Accepted: 27 April 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759245

Digital Object Identifier
doi:10.1214/14-AIHP624

Mathematical Reviews number (MathSciNet)
MR3365978

Zentralblatt MATH identifier
1326.62089

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference

Keywords
Tail-index Kernel estimation Strong uniform consistency

Citation

Goegebeur, Yuri; Guillou, Armelle; Stupfler, Gilles. Uniform asymptotic properties of a nonparametric regression estimator of conditional tails. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 1190--1213. doi:10.1214/14-AIHP624. https://projecteuclid.org/euclid.aihp/1435759245


Export citation

References

  • [1] Y. Aragon, A. Daouia and C. Thomas-Agnan. Nonparametric frontier estimation: A conditional quantile-based approach. Econometric Theory 21 (2) (2005) 358–389.
  • [2] J. Beirlant, Y. Goegebeur, J. Segers and J. Teugels. Statistics of Extremes – Theory and Applications. Wiley Series in Probability and Statistics. Wiley, Chichester, 2004. With contributions from Daniel de Waal and Chris Ferro.
  • [3] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987.
  • [4] V. Chavez-Demoulin and A. C. Davison. Generalized additive modelling of sample extremes. J. R. Stat. Soc. Ser. C. Appl. Stat. 54 (2005) 207–222.
  • [5] H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statistics 23 (4) (1952) 493–507.
  • [6] A. Daouia, L. Gardes and S. Girard. On kernel smoothing for extremal quantile regression. Bernoulli 19 (5B) (2013) 2557–2589.
  • [7] A. Daouia, L. Gardes, S. Girard and A. Lekina. Kernel estimators of extreme level curves. Test 20 (2) (2011) 311–333.
  • [8] A. Daouia and L. Simar. Robust nonparametric estimators of monotone boundaries. J. Multivariate Anal. 96 (2005) 311–331.
  • [9] A. C. Davison and N. I. Ramesh. Local likelihood smoothing of sample extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 (2000) 191–208.
  • [10] A. C. Davison and R. L. Smith. Models for exceedances over high thresholds. J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990) 393–442.
  • [11] U. Einmahl and D. M. Mason. An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 (1) (2000) 1–37.
  • [12] L. Gardes and S. Girard. A moving window approach for nonparametric estimation of the conditional tail index. J. Multivariate Anal. 99 (2008) 2368–2388.
  • [13] L. Gardes and S. Girard. Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels. Extremes 13 (2010) 177–204.
  • [14] L. Gardes and G. Stupfler. Estimation of the conditional tail index using a smoothed local Hill estimator. Extremes 17 (2014) 45–75.
  • [15] S. Girard, A. Guillou and G. Stupfler. Uniform strong consistency of a frontier estimator using kernel regression on high order moments. ESAIM. To appear, 2015. DOI:10.1051/ps/2013050.
  • [16] Y. Goegebeur, A. Guillou and A. Schorgen. Nonparametric regression estimation of conditional tails – the random covariate case. Statistics 48 (2014) 732–755.
  • [17] L. de Haan and A. Ferreira. Extreme Value Theory: An Introduction. Springer, New York, 2006.
  • [18] P. Hall. On some simple estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B Stat. Methodol. 44 (1982) 37–42.
  • [19] P. Hall and N. Tajvidi. Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Statist. Sci. 15 (2000) 153–167.
  • [20] W. Härdle, P. Janssen and R. Serfling. Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16 (1988) 1428–1449.
  • [21] W. Härdle and J. S. Marron. Optimal bandwidth selection in nonparametric regression function estimation. Ann. Statist. 13 (4) (1985) 1465–1481.
  • [22] B. M. Hill. A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 (1975) 1163–1174.
  • [23] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13–30.
  • [24] P. L. Hsu and H. Robbins. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33 (1947) 25–31.
  • [25] M. Lemdani, E. Ould-Said and N. Poulin. Asymptotic properties of a conditional quantile estimator with randomly truncated data. J. Multivariate Anal. 100 (2009) 546–559.
  • [26] Y. P. Mack and B. W. Silverman. Weak and strong uniform consistency of kernel regression estimates. Z. Wahrsch. Verw. Gebiete 61 (1982) 405–415.
  • [27] E. A. Nadaraya. On non-parametric estimates of density functions and regression curves. Theory Probab. Appl. 10 (1965) 186–190.
  • [28] E. Parzen. On estimation of a probability density function and mode. Ann. Math. Statist. 33 (3) (1962) 1065–1076.
  • [29] M. Rosenblatt. Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (3) (1956) 832–837.
  • [30] B. W. Silverman. Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist. 6 (1) (1978) 177–184.
  • [31] R. L. Smith. Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone (with discussion). Statist. Sci. 4 (1989) 367–393.
  • [32] W. Stute. A law of the logarithm for kernel density estimators. Ann. Probab. 10 (1982) 414–422.
  • [33] H. Wang and C. L. Tsai. Tail index regression. J. Amer. Statist. Assoc. 104 (487) (2009) 1233–1240.