Statistics/Probability Theory
Asymptotic distribution of a Pickands-type estimator of the extreme-value index

Presented by: Paul Deheuvels

Gardes, Laurent1; Girard, Stéphane2
1 Université Grenoble 2, LABSAD, B.P. 47, 38040 Grenoble cedex 9, France
2 Université Grenoble 1, LMC-IMAG, B.P. 53, 38041 Grenoble cedex 9, France
Comptes Rendus. Mathématique, Volume 341 (2005) no. 1, pp. 53-58.

One of the main goals of extreme-value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme-value index ξ. The aim of this Note is to establish the asymptotic distribution of the estimator of ξ∈R introduced in Gardes and Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004]. We also give its rate of convergence in some typical situations.

Dans de nombreuses applications, on s'intéresse à l'estimation de probabilités d'évènements rares étant donné un échantillon d'une loi inconnue. Pour ce faire, il est généralement indispensable d'estimer au préalable l'indice des valeurs extrêmes ξ qui caractérise le type de décroissance de la queue de distribution. L'objectif de cette Note est de déterminer la loi asymptotique de l'estimateur de ξ∈R défini dans Gardes et Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004] et d'en déduire sa vitesse de convergence pour certains modèles.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.05.023
Gardes, Laurent 1; Girard, Stéphane 2

1 Université Grenoble 2, LABSAD, B.P. 47, 38040 Grenoble cedex 9, France
2 Université Grenoble 1, LMC-IMAG, B.P. 53, 38041 Grenoble cedex 9, France 
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Gardes, Laurent; Girard, Stéphane. Asymptotic distribution of a Pickands-type estimator of the extreme-value index. Comptes Rendus. Mathématique, Volume 341 (2005) no. 1, pp. 53-58. doi : 10.1016/j.crma.2005.05.023. https://www.numdam.org/articles/10.1016/j.crma.2005.05.023/

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[2] L. Gardes, Estimation d'une fonction quantile extrême, PhD Thesis, Université Montpellier 2, 2003

[3] L. Gardes, S. Girard, A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004

[4] Gnedenko, B. Sur la distribution limite du terme maximum d'une série aléatoire, Ann. Math., Volume 44 (1943), pp. 423-453

[5] Hall, P. On some simple estimates of an exponent of regular variation, J. Roy. Statist. Soc. Ser. B, Volume 44 (1982), pp. 37-42

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

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