Statistics/Probability Theory
Double-thresholded estimator of extreme value index
Comptes Rendus. Mathématique, Volume 337 (2003) no. 4, pp. 287-292.

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).

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).

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00329-7
Gardes, Laurent 1

1 Laboratoire de probabilités et statistique, Université Montpellier 2, place Eugène Bataillon, 34 095 Montpellier cedex 5, France
@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://www.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://www.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://www.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, Volume 337 (2003) no. 4, pp. 287-292. doi : 10.1016/S1631-073X(03)00329-7. http://www.numdam.org/articles/10.1016/S1631-073X(03)00329-7/

[1] Dekkers, A.L.M.; Einmahl, J.H.J.; de Haan, L. A moment estimator for the index of an extreme-value distribution, Ann. Statist., Volume 17 (1989), pp. 1833-1855

[2] Drees, H. Refined Pickands estimator of the extreme value index, Ann. Statist., Volume 23 (1995), pp. 2059-2080

[3] Falk, M. On testing the extreme value index via the POT-method, Ann. Statist., Volume 23 (1995), pp. 2013-2035

[4] Falk, M. 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] Gijbels, I.; Mammen, E.; Park, B.U.; Simar, L. On estimation of monotone and concave frontier functions, J. Amer. Statist. Assoc., Volume 94 (1999), pp. 220-228

[7] Gijbels, I.; Peng, L. Estimation of a support curve via order statistics, Extremes, Volume 3 (1999), pp. 251-277

[8] Hall, P.; Nussbaum, M.; Stern, S.E. On the estimation of a support curve of indeterminate sharpness, J. Multivariate Anal., Volume 62 (1997), pp. 204-232

[9] Härdle, W.; Park, B.U.; Tsybakov, A.B. Estimation of non-sharp boundaries, J. Multivariate Anal., Volume 55 (1995), pp. 205-218

[10] Hill, B.M. A simple general approach to inference about the tail of a distribution, Ann. Statist., Volume 3 (1975), pp. 1163-1174

[11] Marohn, F. Testing the Gumbel hypothesis via the P.O.T. method, Extremes, Volume 1 (1998) no. 2, pp. 191-213

[12] Marohn, F. Local asymptotic normality of truncated models, Statist. Decisions, Volume 17 (1999), pp. 237-253

[13] Pickands, J. III Statistical inference using extreme-order statistics, Ann. Statist., Volume 3 (1975), pp. 119-131

[14] Resnick, S.I. Extreme Values, Regular Variation, and Point Process, Springer-Verlag, New York, 1987

Cited by Sources: