no. 3
Uniform asymptotic properties of a nonparametric regression estimator of conditional tails
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1190-1213.

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.

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.

DOI : 10.1214/14-AIHP624
Mots clés : tail-index, kernel estimation, strong uniform consistency 
@article{AIHPB_2015__51_3_1190_0,
     author = {Goegebeur, Yuri and Guillou, Armelle and Stupfler, Gilles},
     title = {Uniform asymptotic properties of a nonparametric regression estimator of conditional tails},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1190--1213},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {3},
     year = {2015},
     doi = {10.1214/14-AIHP624},
     mrnumber = {3365978},
     zbl = {1326.62089},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/14-AIHP624/}
}
TY  - JOUR
AU  - Goegebeur, Yuri
AU  - Guillou, Armelle
AU  - Stupfler, Gilles
TI  - Uniform asymptotic properties of a nonparametric regression estimator of conditional tails
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 1190
EP  - 1213
VL  - 51
IS  - 3
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/14-AIHP624/
DO  - 10.1214/14-AIHP624
LA  - en
ID  - AIHPB_2015__51_3_1190_0
ER  - 
%0 Journal Article
%A Goegebeur, Yuri
%A Guillou, Armelle
%A Stupfler, Gilles
%T Uniform asymptotic properties of a nonparametric regression estimator of conditional tails
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 1190-1213
%V 51
%N 3
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/14-AIHP624/
%R 10.1214/14-AIHP624
%G en
%F AIHPB_2015__51_3_1190_0
Goegebeur, Yuri; Guillou, Armelle; Stupfler, Gilles. Uniform asymptotic properties of a nonparametric regression estimator of conditional tails. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1190-1213. doi : 10.1214/14-AIHP624. http://archive.numdam.org/articles/10.1214/14-AIHP624/

[1] Y. Aragon, A. Daouia and C. Thomas-Agnan. Nonparametric frontier estimation: A conditional quantile-based approach. Econometric Theory 21 (2) (2005) 358–389. | MR | Zbl

[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. | MR | Zbl

[3] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1987. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | MR | Zbl

[6] A. Daouia, L. Gardes and S. Girard. On kernel smoothing for extremal quantile regression. Bernoulli 19 (5B) (2013) 2557–2589. | MR | Zbl

[7] A. Daouia, L. Gardes, S. Girard and A. Lekina. Kernel estimators of extreme level curves. Test 20 (2) (2011) 311–333. | MR | Zbl

[8] A. Daouia and L. Simar. Robust nonparametric estimators of monotone boundaries. J. Multivariate Anal. 96 (2005) 311–331. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[14] L. Gardes and G. Stupfler. Estimation of the conditional tail index using a smoothed local Hill estimator. Extremes 17 (2014) 45–75. | DOI | MR | Zbl

[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. | MR | Zbl

[16] Y. Goegebeur, A. Guillou and A. Schorgen. Nonparametric regression estimation of conditional tails – the random covariate case. Statistics 48 (2014) 732–755. | DOI | MR | Zbl

[17] L. De Haan and A. Ferreira. Extreme Value Theory: An Introduction. Springer, New York, 2006. | MR | Zbl

[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. | MR | Zbl

[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. | DOI | MR

[20] W. Härdle, P. Janssen and R. Serfling. Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16 (1988) 1428–1449. | DOI | MR | Zbl

[21] W. Härdle and J. S. Marron. Optimal bandwidth selection in nonparametric regression function estimation. Ann. Statist. 13 (4) (1985) 1465–1481. | MR | Zbl

[22] B. M. Hill. A simple general approach to inference about the tail of a distribution. Ann. Statist. 3 (1975) 1163–1174. | MR | Zbl

[23] W. Hoeffding. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13–30. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[27] E. A. Nadaraya. On non-parametric estimates of density functions and regression curves. Theory Probab. Appl. 10 (1965) 186–190. | DOI | MR | Zbl

[28] E. Parzen. On estimation of a probability density function and mode. Ann. Math. Statist. 33 (3) (1962) 1065–1076. | MR | Zbl

[29] M. Rosenblatt. Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (3) (1956) 832–837. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[32] W. Stute. A law of the logarithm for kernel density estimators. Ann. Probab. 10 (1982) 414–422. | DOI | MR | Zbl

[33] H. Wang and C. L. Tsai. Tail index regression. J. Amer. Statist. Assoc. 104 (487) (2009) 1233–1240. | MR | Zbl

Cité par Sources :