[Un estimateur de l'indice des valeurs extrêmes à double seuil]
Dans cette Note, nous proposons un estimateur de l'indice des valeurs extrêmes construit en utilisant uniquement le nombre de points qui dépassent des seuils aléatoires. On démontre qu'il est faiblement consistant et asymptotiquement normal. Du résultat de convergence en loi, on déduit que la vitesse de convergence de notre estimateur est une puissance de la taille de l'échantillon. A notre connaissance, cette vitesse n'est atteinte par aucun autre estimateur de l'indice des valeurs extrêmes. A l'aide de simulations, nous comparons notre estimateur à l'estimateur des moments (Dekkers et al., Ann. Statist. 17 (1989) 1833–1855).
The purpose of this Note is to propose an estimator of the extreme value index constructed by using only the number of points exceeding random thresholds. We prove the weak consistency and the asymptotic normality of this estimator. We deduce from this last result that the rate of convergence of our estimator is in a power of the sample size. To our knowledge, this rate of convergence is not reached by any other estimate of the extreme value index. Through a simulation, we compare our estimator to the moment estimator (Dekkers et al., Ann. Statist. 17 (1989) 1833–1855).
Accepté le :
Publié le :
@article{CRMATH_2003__337_4_287_0,
author = {Gardes, Laurent},
title = {Double-thresholded estimator of extreme value index},
journal = {Comptes Rendus. Math\'ematique},
pages = {287--292},
publisher = {Elsevier},
volume = {337},
number = {4},
year = {2003},
doi = {10.1016/S1631-073X(03)00329-7},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(03)00329-7/}
}
TY - JOUR AU - Gardes, Laurent TI - Double-thresholded estimator of extreme value index JO - Comptes Rendus. Mathématique PY - 2003 SP - 287 EP - 292 VL - 337 IS - 4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(03)00329-7/ DO - 10.1016/S1631-073X(03)00329-7 LA - en ID - CRMATH_2003__337_4_287_0 ER -
%0 Journal Article %A Gardes, Laurent %T Double-thresholded estimator of extreme value index %J Comptes Rendus. Mathématique %D 2003 %P 287-292 %V 337 %N 4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(03)00329-7/ %R 10.1016/S1631-073X(03)00329-7 %G en %F CRMATH_2003__337_4_287_0
Gardes, Laurent. Double-thresholded estimator of extreme value index. Comptes Rendus. Mathématique, Tome 337 (2003) no. 4, pp. 287-292. doi : 10.1016/S1631-073X(03)00329-7. http://archive.numdam.org/articles/10.1016/S1631-073X(03)00329-7/
[1] A moment estimator for the index of an extreme-value distribution, Ann. Statist., Volume 17 (1989), pp. 1833-1855
[2] Refined Pickands estimator of the extreme value index, Ann. Statist., Volume 23 (1995), pp. 2059-2080
[3] On testing the extreme value index via the POT-method, Ann. Statist., Volume 23 (1995), pp. 2013-2035
[4] Local asymptotic normality of truncated empirical processes, Ann. Statist., Volume 26 (1998), pp. 692-718
[5] L. Gardes, Estimation de l'indice de valeur extrême, Rapport de Recherche ENSAM-INRA-UM2 02-06, 2002
[6] On estimation of monotone and concave frontier functions, J. Amer. Statist. Assoc., Volume 94 (1999), pp. 220-228
[7] Estimation of a support curve via order statistics, Extremes, Volume 3 (1999), pp. 251-277
[8] On the estimation of a support curve of indeterminate sharpness, J. Multivariate Anal., Volume 62 (1997), pp. 204-232
[9] Estimation of non-sharp boundaries, J. Multivariate Anal., Volume 55 (1995), pp. 205-218
[10] A simple general approach to inference about the tail of a distribution, Ann. Statist., Volume 3 (1975), pp. 1163-1174
[11] Testing the Gumbel hypothesis via the P.O.T. method, Extremes, Volume 1 (1998) no. 2, pp. 191-213
[12] Local asymptotic normality of truncated models, Statist. Decisions, Volume 17 (1999), pp. 237-253
[13] Statistical inference using extreme-order statistics, Ann. Statist., Volume 3 (1975), pp. 119-131
[14] Extreme Values, Regular Variation, and Point Process, Springer-Verlag, New York, 1987
Cité par Sources :
